Detecting outliers through boxplots of the features

This function detects univariate outliers simultaneously using boxplots
of the features:

require(dprep)

data(diabetes)

outbox(diabetes,nclass=1)

box plots online

B.O.D.M.A.S...please tell me u remember?

Okay, I know I should keep this blogging thing consistent but I actually have to pay attention to biostatistics sometimes. Speaking of this class, the other day I waltzed in, late as usual to get my exam results...not bad..not bad at all. Apparently, not everyone did well in the exam, people couldn't compute basic math and hence the tantrum that my instructor was throwing that morning.
So I walk in and take my exam and as i'm perusing thru, he starts...I'm very dissapointed in some of you, I was shocked to see that people couldn't do basic math, this messed up their entire flowcharts and histograms..let's review..

19+22-40/52+rt16-5 is equal to what?

The teacher asks what the process was and since most of the people in my class went to elementary here they have this whole thing about Mary's mother and grandmother doing something..and I'm like what?? I don't know what the hell ure talking about...so everyone in class is shocked wondering how I got ! my ass into college and didn't go through elementary.

So the teacher turns to me and asks just how I did the freaking problem if I didn't know the basic ruless. My response: BODMAS....yaani the one mrs.Gitonga taught me..if you don't know...

• Brackets
• Of
• Division
• Multiplication
• Addition
• Subtraction

The man looks at me and doesn't really understand what the crap I'm talking about. So I get up and head to the board and explain myself while doing the freaking problem. Most of these punks are still lost and so it takes another 20 mins before everyone understands what the hell I'm talking about.

Response: Wow...u learnt that from Africa? or did u read it on the internet. My response...Mary and her grandmother didn't do very well in teaching you math.

that's my piece.

bodmas problems

Matrix Algebra for Dummies

A free read on basic matrix algebra.

Also some video lessons here.

algebra for dummies free online

Trigonometric Identities - Basic Identities

trigonometric identities are specific equalities that express one trig function in terms of other trig functions. They are fairly straightforward, but they take some work to derive them. If you are comfortable with simple derivations, you shouldn't have any problems though. Personally, I find it easier to remember the basic set of identities, and derive the more complex ones from those, rather than trying to memorize all of them... although some people are more comfortable just to memorize them.

The basic identities are traditionally visualized with a triangle formed by a radius r, length x, and height y:

The basic trig definitions can easily be seen:
Sin(theta) = y/r..... opposite/hypotenuse
Cos(theta) = x/r..... adjacent/hypotenuse
Tan(theta) = y/x..... opposite/adjacent

If we now apply the Theorem of Pythagoras, we can see:

r^2 = x^2 + y^2

Dividing everything by r^2 gives:

1 = (x^2)/(r^2) + (y^2)/(r^2)
1 = (x/r)^2 + (y/r)^2

And then, subbing in the basic definitions, we get:

1 = [Cos(theta)]^2 + [Sin(theta)]^2

And that is the first basic identity. Nothing to it. It's proper name is! the Pythagorean Trigonometric Identity. I'll rewrite it in p! roper no tation to clean it up a bit... (Blogger is a pain with superscripts and fonts)

Another basic relationship starts with:

Tan(theta) = y/x

But, then sub in the Sine and Cosine definitions (isolated for x and y, respectively) to give

Tan(theta) = (r*Sin(theta)) / (r*Cos(theta))
Tan(theta) = Sin(theta) / Cos(theta)

And that's it again. This is call! ed the Ratio Identity:

Those are now two of the simplest trig identities from which most of the others can be derived.

trigonometric identities

Algebra: Surds

Surds are numbers left in 'square root form' (or 'cube root form' etc). They are therefore irrational numbers. The reason we leave them as surds is because in decimal form they would go on forever and so this is a very clumsy way of writing them. (See here for notes and examples )

Skills needed to answer GCSE questions

Using Surds in calculations

Simplifing Surds

Rationalise Surds (which needs you to remember the difference of 2 squares)

Helpful resources, examples and links

	BBC Bitesize is always good GCSE Maths tutor - technical but some worksheets and videos
Some people fing watching a YouTube video can help, but be aware that they use real Maths teachers
MyMaths Surds1 goes through the topics in a gentle way.
Surds2 - a little harder

surds maths

Kyle Bobby Dunn - Rural Route No. 2 (Standard Form, 2010)

Kyle Bobby Dunn - Dissonant Distances


Standard Form put out a limited edition 3" CD-R from the new master of minimal drone, Kyle Bobby Dunn. Only two songs, "Dissonant Distances" & "Senium III," but they're both around the 10 minute mark and are as gorgeous as can be. Of course "Dissonant Distances" has its moments of, well, dissonance, using a strange sort of muted industrial ambience. It's a bit more textured than his previous stuff, a little more static & crackle as opposed to smothered strings. Not what I was expecting and in a very awesome way.

The second track is the slow moving Stars Of The Lid style drone you've come to know & love. Soft, delicate, and smooth, anything but ambient backgrou! nd sounds. The layers of beauty suck you in & hypnotize you. L! ovely to nes emanating off the golden lakes in the clouds. Seriously special stuff.

I almost hope Dunn isn't going the route of Aidan Baker, putting out 30 releases every year because honestly, I can't do that with another artist who's this good. It's bound to either empty my wallet or put me in a deep funk knowing I can't have it all. Most likely the former.

What is Standard Form

Converting Point-Slope Form to Standard Form

I previously described how to obtain the equation of a line, and how to express that in both point-slope form and standard form. While both equations describe the exact same line, sometimes you may be asked to express the line in a specific way, and you need to be able to manipulate and rearrange the provided equation to make it look like the other form. I will show an example of how this can be done.

Reminders (refer to the posts linked above for more details)

Point-slope looks like this:
(y-y1) = m(x-x1), which is the general way of saying y=mx+b

Standard form looks like this:
Ax + By = C

Example: Express the equation y=5x-10 in standard form. State the values for A, B, and C.

Basically, what you want to do is move all the x and y terms over to one side, and move the constants (terms with no variables) over to the other. Combine and simplify where possible. That's all there is to it. "A" will be the term left over in front of x, "B" will be with y, and C will be the value not attached to a variable.

y=5x-10

10=5x-y

So:
5x-y=10
A=5, B=(-1), C=10
(remember the standard form has a "+", so a "-" in your answer implies a coefficient of (-1).

Let's try another one:

Example:Express the equation y=(4/ 3)x+2 in standard form. State the values for A, B, and C.

This one works the same way, but there is something else that can be done, as I will demonstrate.

y=(4/3)x+2
(-2)=(4/3)x-y
So:
(4/3)x-y=(-2)
A=(4/3), B=(-1), C=(-2)
There is nothing wrong with this answer. It is properly rearranged, and the coefficients have been stated. However, usually it is a good idea to not have fractions (ie. have nothing in the denominator). So, to do this, you work our final answer a bit further, so that all the values are in the numerators.

(4/3)x-y=(-2)
Multiply all terms by 3, to remove it from the denominator of the first term. This gives:
4x-3y=(-6)
A=4, B=(-3), C=(-6)

Again, this answer describes the exact same line as the initial answer without the extra moves, so technicall! y, they are both right. It is just a common convention to keep things in the numerator wherever possible.

Converting from the Standard Form to the Point-slope form is basically just the reverse. Try it for yourself with these examples!

What is Standard Form

Homework

Grade 8: Multiplying Special Cases: Page 391-392 all even numbered problems ( 2- 56)

Grade 7: Two-Step Equations with Fractions and Decimals.  As you solve these problems try to solve them by clearing the equation  of fractions and decimals.  Work on pages 356-357 problems 2-56 evens.

Grade 6: Note Taking. I will check your Math Notebooks for the note on lesson 7-4 Quadrilateral and Other Polygons.  Please make sure you draw the figures, include vocabulary terms, and do all California Check Questions.

Solving two step equations with fractions

Snap Game for Factoring Trinomials

This game of Snap is one that your students can play on the computer to practice factoring trinomials. One card displaying a quadratic equation is shown, and the other pile of cards is made up of factors of quadratic equations. Cards from this pile are flipped over. You "snap" when the factors from a card in this pile match up to the quadratic equation shown. This would be easy to make up several  decks of cards for different students to play on their own for extra practice and a little fun competition.

Solve quadratic equations by factoring

Relativistic Quantum Mechanics: The Klein-Gordon Equation

The Schrödinger equation, used in non-relativistic quantum mechanics, is not Lorentz invariant. This can be seen upon inspection since it is first order in time derivatives, and second order in spatial derivatives. Special relativity must treat space and time on an equal footing, i.e. they must be of the same order. Here's the Schrödinger equation for reference:

i\hbar\frac{\partial\psi}{\partial t} = -\frac{\hbar^2}{2m}\nabla^2\psi + V(\mathbf{r})\psi

In order to treat quantum mechanics in a Lorentz invariant framework, we must begin with the relativistic energy-momentum relationship:

E^2 = p^2c^2 + m_0^2c^4

Quantum mechanical operators for the energy and momentum can be defined as:

\begin{align*}
\hat{E}=-i\hbar\frac{\partial}{\partial t} \\
\hat{p}=-i\hbar\nabla
\end{align*}

For a plane wave solution of the for! m

\psi(\mathbf{x},t)=A\exp(i\mathbf{k}\cdot\mathbf{x}-i\omega t)

, these operators give the expected relationships for energy and momentum as their eigenvalues:

\begin{align*}
\hat{E}\psi = -i\hbar i\omega\psi = \hbar\omega\psi \\
\hat{p}\psi = -i\hbar i\mathbf{k} = \hbar\mathbf{k}
\end{align*}

These operators (squared) can be substituted directly into the relativistic energy-momentum relationship. From now on, I'm going to simplify matters by using natural units

(\hbar=c=1)

and writing the rest mass

m_0=m

.

\begin{align*}
\hat{E}^2\psi = \hat{p}^2\psi+m^2\psi\\
\Rightarrow -\frac{\partial^2}{\partial t^2}\psi = -\nabla^2\psi + m^2\psi \\
\Rightarrow \left(-\frac{\partial^2}{\partial t^2} + \nabla^2 -m^2\right)\ps!  i = 0 \\
\Rightarrow \left(\partial_\mu\partial^\mu + m^2!  \right)\ psi = 0 \\
\Rightarrow \left( \Box +m^2\right)\psi(\mathbf{x},t) = 0
\end{align*}

Where

\Box = \partial_\mu\partial^\mu = \frac{\partial^2}{\partial t^2}-\nabla^2

.
This is the Klein-Gordon Equation, and has plane-wave solutions of the form:

\begin{align*}
\psi(\mathbf{x},t)=N\exp(-iEt+i\mathbf{p}\cdot\mathbf{x}) \\
= N\exp(-ip\cdot x) \\
\mathrm{where~}p\cdot x = p_\mu x^\mu = Et - \mathbf{p}\cdot\mathbf{x}
\end{align*}

Since

E^2=p^2+m^2

, the energy solutions are:

E = \pm\sqrt{\mathbf{p}^2+m^2}

i.e. there are positive and negative energy states associated with the Klein-Gordon equation. Before discussing the meaning of these, I'd like to first obtain the probability current for the Klein! -Gordon equation. For the Schrödinger equation, the probability density is the square of the wavefunction,

\rho = |\psi|^2 = \psi\psi^\dagger

. If we take the Klein-Gordon equation as:

\frac{\partial^2\psi}{\partial t^2}-\nabla^2\psi + m^2\psi = 0

We can multiply by

\psi^\star

and subtract

\psi

the complex conjugate of the Klein-Gordon equation. This results in:

\frac{\partial\rho}{\partial t}+\nabla\cdot\mathbf{j}=0

Where:

\begin{align*}
\rho=i\left[\psi^\star \frac{\partial\psi}{\partial t}-\left(\frac{\partial\psi^\star}{\partial t}\right)\psi\right] \\
\mathbf{j} = \frac{1}{i}\left[\psi^\star\nabla\psi - (\nabla\psi^\star)\psi\right]
\end{align*}

In four-vect! or notation, one can write the above as:

\partial_\mu j^\mu = 0

with

j^\mu = (\rho, \mathbf{j}) = i\left[\psi^\star\partial^\mu\psi - (\partial^\mu\psi^\star)\psi\right]

Since

\psi

is Lorentz invariant, and

\partial^\mu

is a contravariant four-vector,

j^\mu

is also contravariant.

The spatial current j is identical to the Schrödinger current, but for the Klein-Gordon equation, the probability density contains time derivatives since K.G. is second-order in time derivatives. The probability density

\rho

is not, therefore, constrained to be positive definite. For plane wave solutions:

\begin{align*}
\psi=N\exp(-iEt-i\mathbf{p}\cdot\mathbf{x}) \\
\rho!   = 2|N|^2E \\
E = \pm\sqrt{\mathbf{p}^2+m^2} \\
\Rightarrow \rho = \pm 2|N|^2\sqrt{\mathbf{p}^2+m^2}
\end{align*}

The probability density is therefore positive for positive-energy states and negative for negative energy states. The Klein-Gordon equation was abandoned for some time as a result of this, but the negative energy states may be interpreted as representing particles moving backwards in time, corresponding to anti-particles moving forwards in time! (The Feynman interpretation). The Klein-Gordon equation describes the relativistic quantum mechanics of (massless) spin-zero particles. (And no such fundamental particles exist within the Standard Model, but the neutral pion, composed of

\frac{1}{\sqrt{2}}(u\bar{u}-d\bar{d})

can be considered at low energies. In order to deal with fermions (spin-1/2 particles) we need to look at the Dirac equation, which I might write about another time.

probability equation

What is a Probability?

Now that we have somewhat discussed the idea of a state, we can proceed to discuss the next building block for understanding the idea of a quantum state. As quantum states are rather intrinsically tied to the notion of a probability distribution, it is worth talking about classical probability for a few minutes. I won't try to teach all of statistics, or even necessarily the most interesting parts, so much as I will try to communicate a bit about the subset of probability theory that gets used in quantum information.

Imagine that you have a coin. If we plan on using this coin to decide something of value to us, such as the order of play in some game, then we are naturally going to be interested in whether this coin is fair. To do so, we need to understand what we mean by the word "fair" here. One way of defining fair may be to say that no one can predict the outcome of a flip of the coin in any way that is more reliable than just guessing. That is, that there is no information that an observer could have which would give them a leg up on coin prediction. This is a notion which we will return to later.

Another way to capture our intuition about fairness, which turns out to be easier to discuss at first, is to imagine that the coin is flipped a very large number of times. We would then demand that a fair coin give approximately as many "heads" as "tails," so that neither player in a coin flipping game is advantaged over the other.

We can formalize this second notion by defining the fraction p of the coin flips which come up heads and then demanding that p = 0.5. We then say that p  is the probability of obtaining a heads from our coin flip experiment. If the coin is unfair, then we can imagine p taking on any value from 0 to 1 (there can't very well be less than zero heads, nor more heads than flips of the coin). In either case, however, our notion of probability as being a proportion of experiments having a specific outcome depends on considering very large numbers of trials, such that we can mostly ignore unlikely coincidences such as five heads in a row.


Of course, we needn't restrict ourselves to coins. More generally, we can think of a source as being some device or system that produces a series of events drawn from a list of all possible events. Under this view, the fair coin was a source that produced events from the list {heads, tails}, but we can have any number (well, any number in â„•, anyway) of possible events in such a list. Then, we will have a probability distribution function, written p(x), that tells us the fraction of experiments in which a source X produces the event x. For example, in the coin example, p = p(heads). Of course, summing this function over all possible events must give probability 1, since in all experiments, something happens. Written in symbols:

Once we have this probability distribution function, we can do some very nifty things. The primary one that I wish to talk about here is how to connect the idea of a probability distribution to the earlier intuition about uncertainty. To do this, we exploit the Shannon entropy-- I won't derive it here, but it turns out that the number of bits required to describe (on average) the outcome of an experiment with a particular source, written H(p), is given by a nice and compact formula:

Here, by lg x, we mean the base-2 logarithm of x, which can be obtained by the change of base formula lg x = log x / log 2.

A quick check with a calculator tells us that in the fair coin example, H(p) = 1, which we would expect: no observer can do better at recording coin flips than to simply write down each coin flip as it occurs, using a 0 for heads and a 1 for tails (or vice versa). Another quick check tells us that if we have an unfair coin with p(heads) = 0.9, we get H ≈ 0.46 bits/flip, meaning that someone who knows that the coin is unfair is much less uncertain of the outcome of a coin flip than someone who doesn't. This example shows a close relation between the concepts of probability, uncertainty and entropy, as we shall see more of in the future.

For now, however, I hope you enjoyed a bit of introduction to and philosophizing about probabilities!

what is probability

Pre-Algebra Lesson Plan (Reteach)

February 2-6, 2009
RETEACH PRE-ALGEBRA THIS WEEK; REVIEW DC BAS QUESTIONS

MONDAY
2.2.2009

6PRA1 Use the properties of equality to solve problems using letter name variables (eg ¼ + x = 7/12)

WARM UP:
Give the definition:
-Concepts
-Problems
-variables
- equality
-inequality

The student will:
T1 GOAL: Distinguish between variables and numbers in an equation.
T2 GOAL: Solve an equation with a letter name variable as the solution (eg, 4+1=x)
T3 GOAL: Solve an equation with a letter name variable in a position other than the solution (eg, 9+x=10)

By answering at least 4/5 math problems on the board and on p! aper.

ESSENTIAL QUESTION:
How do we solve problems of inequality using letter name variables?

WHOLE GROUP INSTRUCTION:
Show this PowerPoint presentation on Linear Equations: http://jc-schools.net/PPT/linear-equations.ppt

SMALL GROUP INSTRUCTION: Break the students in three groups and give them 5 math problems involving linear equations. Ask them to:

T1 GOAL: Distinguish between variables and numbers in an equation.
T2 GOAL: Solve an equation with a letter name variable as the solution (eg, 4+1=x)
T3 GOAL: Solve an equation with a letter name variable in a position other than the solution (eg, 9+x=10)

ASSESSMENT:
Online game:
http://www.ies.co.jp/math/java/geo/linf/lin! f.html

TUESDAY
2.3.2009

6PRA1 Use the properties of equality to solve problems using letter name variables
(eg ¼ + x = 7/12)

WARM UP:
Page 1, #s 1-3http://glencoe.mcgraw-hill.com/sites/dl/free/0078884802/633197/alg1sgi.pdf

The student will:
T1 GOAL: Distinguish between variables and numbers in an equation.
T2 GOAL: Solve an equation with a letter name variable as the solution (eg, 4+1=x)
T3 GOAL: Solve an equation with a letter name variable in a position other than the solution (eg, 9+x=10)

By answering at least 4/5 math problems on the board and on paper.

ESSENTIAL QUESTION:How do we solve problems of inequality using letter nam! e variables?

WHOLE GROUP INSTRUCTION:
Show this PowerPoint presentation on Properties of Equality http://www.cohs.com/teachers/docs/50_Associative%20Property%20of%20Addition%20etc.ppt#1

SMALL GROUP INSTRUCTION: Break the students in three groups and give them 5 math problems involving equalities. (http://glencoe.mcgraw-hill.com/sites/dl/free/0078884802/633197/alg1sgi.pdf) Ask them to finish answer page 1 # 7-11:

T1 GOAL: Distinguish between variables and numbers in an equation.
T2 GOAL: Solve an equation with a letter name variable as the solution (eg, 4+1=x)
T3 GOAL: Solve an equation with a letter name variable in a position other than the solution (eg, 9+x=10)

ASSESSMENT:Online game:http://www.ies.co.jp/math/java/geo/linf/linf.html

WEDNESDAY
2.4.2009

6PRA4 Simplify expressions of the first degree by combining like terms and using specific values.

WARM UP:
Page 1, #s 4-6http://glencoe.mcgraw-hill.com/sites/dl/free/0078884802/633197/alg1sgi.pdf

The student will:
T1 GOAL: Distinguish between an equation and an expression.
T2 GOAL: Label a problem as equal or unequal
T3 GOAL: Match two equations that are equal

By solving linear problems on the Promethean Board.

ESSENTIAL QUESTION:
How do we know which are like terms?

WHOLE GROUP IN! STRUCTION:
Show this PowerPoint presentation on Linear Equations http://jc-schools.net/PPT/linear-equations.ppt

SMALL GROUP INSTRUCTION: Break the students in three groups and give them 5 math problems involving equalities. (http://glencoe.mcgraw-hill.com/sites/dl/free/0078884802/633197/alg1sgi.pdf) Ask them to finish answer page 1 # 12-16

T1 GOAL: Distinguish between an equation and an expression.
T2 GOAL: Label a problem as equal or unequal
T3 GOAL: Match two equations that are equal

ASSESSMENT:
Online game:http://www.ies.co.jp/math/java/geo/linf/linf.html

THURSDAY
2.5.2009
6PRA4 S implify expressions of the first degree by combining like terms and using specific values.

WARM UP:
Page 2, #s 1-3http://glencoe.mcgraw-hill.com/sites/dl/free/0078884802/633197/alg1sgi.pdf

The student will:
T1 GOAL: Distinguish between an equation and an expression.
T2 GOAL: Label a problem as equal or unequal
T3 GOAL: Match two equations that are equal

By solving linear problems on the Promethean Board.

ESSENTIAL QUESTION:
How do we know which are like terms?

WHOLE GROUP INSTRUCTION:
Show this PowerPoint presentation on Linear Equations http://jc-schools.net/PPT/linear-equations.ppt

SMALL GROUP INSTRUCTION: Break the students in three groups and give them 5 math p! roblems involving equalities. (http://glencoe.mcgraw-hill.com/sites/dl/free/0078884802/633197/alg1sgi.pdf) Ask them to finish answer page 2 # 7-12

T1 GOAL: Distinguish between an equation and an expression.
T2 GOAL: Label a problem as equal or unequal
T3 GOAL: Match two equations that are equal

ASSESSMENT:
Online game:http://www.ies.co.jp/math/java/geo/linf/linf.html

FRIDAY
2.6.2009

6PRA4 Simplify expressions of the first degree by combining like terms and using specific values.


WARM UP:
Page 2, #s 4-6
http://glencoe.mcgraw-hill.com/sites/dl/free/0078884802/633197/alg1sgi.pdf

The student will:
T1 GOAL: Distinguish between an equation and an expression.
T2 GOAL: Label a problem as equal or unequal
T3 GOAL: Match two equations that are equal

By solving linear problems on the Promethean Board.

ESSENTIAL QUESTION:
How do we know which are like terms?

FRIDAY FUNDAY!

Pre-Algebra Jeopardy Game:
http://schoolcenter.brentwood.k12.ny.us/education/components/docmgr/default.php?sectiondetailid=7017&fileitem=371&catfilter=85&PHPSESSID=2b87035eb2904fbfa0d3b36c78b2c4a1

WHOLE GROU! P INSTRUCTION:
Explain the mechanics of the game

SMALL GROUP INSTRUCTION:Break the class into three teams, play the jeopardy game!

ASSESSMENT:
Online game:http://www.ies.co.jp/math/java/geo/linf/linf.html

HOMEWORK:
Access Code: YQ5192

Solving pre algebra equations

Math textbook reviews

It is quite difficult to navigate around the Mathematically Correct site when looking for specific math textbook reviews. There is no search function. Updating is also sporadic and it is not possible to tell at a glance which entries are new or the date of other entries. Important articles like Barry Garelick's Miracle Math and An A-Maze-ing Approach To Math are not linked (at least I haven't been able to find links).

I did find a page that puts together reviews of second, fifth and seventh grade math textbooks in a handy fashion.

Glencoe/McGraw-Hill Pre-Algebra, an Integrated Transition to Algebra and Geometry comes out on top for seventh grade in these reviews. On the other hand! , amazon reviewers panned this textbook. Where is the truth?

I use Scott Foresman/Addison Wesley Middle School Math and am quite happy with it.

Dale Seymour Publications Connected Mathematics Program gets an F from mathematically correct. CMP is a cult item in many districts, including here in Chicago.

It's worth citing the review for fun. I particularly like phrases like these: "If this topic is covered, it is extremely difficult to find." Also this: "Fractions [1.0] This topic is missing or cryptic."

Mathematically Correct Seventh Grade Mathematics Review
Dale Seymour Publications
Connected Mathematics Program
Lappan, Fey, Fitzgerald, Friel and Phillips
Menlo Park


--------------------------------------------------------------------------------


Introduction

This is part of a series of second, fifth! , and seventh grade Mathematics Program Reviews. This review i! ncludes a summary of the structure of the program, evaluations of a selected set of content areas, and evaluations of program quality. Ratings in these areas were made on a scale from 1 (poor) to 5 (outstanding). The overall evaluation was made using the traditional system of letter grades. For details of the methods used in this evaluation see Methods for Seventh Grade Program Reviews.

Student Text Structure

This course is composed of 8 "books" of about 70-90 pages each.

Each book is arranged around a mathematical topic

1. Variables and patterns
2. Stretching and shrinking
3. Comparing and scaling
4. Accentuate the negative
5. Moving straight ahead
6. Filling and wrapping
7. What do you expect?
8. Data around us

Content Area Evaluations

Properties, Order of Operations [1.0]

If this topic is covered, it is extremely difficult to find. In any case, the coverage is insuffici! ent.

Exponents, squares, roots [1.0]

This topic is very weak. Positive integers are raised to whole number powers only in the context of prime factorization. The small coverage of scientific notation includes only positive integer exponents and heavily emphasizes the use of calculators. All other topics are completely absent.

Fractions [1.0]

This topic is missing or cryptic.

Decimals [1.0]

None of these topics are presented in this book. Expressing decimals as percents is assumed. Perhaps students were taught to make this conversion on their calculators in an earlier grade.

Percents [1.2]

There is no evidence of development of this topic in this book. At the start of book 3 "percent" is described as one of the "terms developed in previous units." Perhaps so, but if so, the level of development was low. There is certainly no teaching of percent. Some percent problems are intermixed with r! atio problems in the various exercises, but there is no instru! ction on interconverting fractions, decimals and percents. There are almost no word problems on discount, markups, commissions, increase or decrease. Some "scale factors" for similarity are expressed as percents.

Proportions [3.5]

This is an adequate treatment of this topic. Most of the grade 7 topics are covered at some level. It is interesting that although proportions are used relatively often, by the standards of this book, they are not referred to as such as the authors have decided to put "proportion" in the list of nonessential terms at the front of the book.

Expressions and Equations - Simplifying and Solving [1.1]

This book is devoid of algebraic manipulation. The only solving of equations is graphical, and that is limited to problems involving direct variation. It is difficult to see how a student can become prepared for any mathematics-based profession with this little instruction in the key skills leading to algebra.

Expressions and Equations - Writing [2.1]

This book has a heavy emphasis on using proportions to find the lengths of similar parts of similar figures. On the other hand, there is very little practice or instruction in writing equations from a verbal description. The problems related to this topic are stretched over an excessive period of time as students answer endless questions about the situation. Essentially all of these problems are dealt with via tables and then in a graphical context.

Graphing [4.0]

The one advantage of the heavy emphasis that this book places on graphical over analytic solutions is that graphing is covered moderately well. Nearly all topics that should be covered are covered.

Shapes, Objects, Angles, Similarity, Congruence [2.5]

Formulas and derivations, or even "discoveries" of area of two dimensional figures are not given. They may be assumed to have been mastered at an earlier year. Surface a! rea and volume are "discovered" in a long series of constructi! on proje cts, many of which look doomed to failure. At the end of this formulas that have been discovered are not explicitly stated in the text. The teacher's manual suggests that the appropriate formulae will "come out" in discussion. If the student discovered the wrong formula, or forgot to write it down, good luck, since there is no way to look back and remember a formula.

The exercises on finding volumes of irregular objects using displacement are interesting extensions. On the other hand, much of the teaching is absurd and again abjures analysis for experiment. For example, students spend who-knows-how-much-time filling cylinders and cones with beans to discover, approximately, the relationship between the volume of cylinders and the volume of cones. This is a waste of time and inaccurate. Unfortunately, this could describe many of the activities in this book.

Area, Volume, Perimeter, Distance [1.0]

Essentially none of the grade 7 level topics ar! e covered. They may be assumed from previous years, but one cannot assume that from the presentation.

Program Quality Evaluations

Mathematical Depth [1.7]

There is very little mathematical content in this book. Students leaving this course will have no background in or facility with analytic or pre-algebra skills.

Quality of Presentation [1.4]

This book is completely dedicated to a constructivist philosophy of learning, with heavy emphasis on discovery exercises and rejection of whole class teacher directed instruction. The introduction to Part 1 says "Connected Mathematics was developed with the belief that calculators should be available and that students should decide when to use them." In one of the great understatements, the Guide to the Connected Mathematics Curriculum states, "Students may not do as well on standardized tests assessing computational skills as students in classes that spend" time practicing these ! skills.

Quality of Student Work [1.5]

St! udents a re busy, but they are not productively busy. Most of their time is directed away from true understanding and useful skills.

Overall Program Evaluation

F
Overall Evaluation [1.7]

This rating is perhaps deceivingly high, as 7 of the 11 topics rate no higher than 1.2. The rating is as high as it is based largely on two high subscores, proportions and graphing. It is impossible to recommend a book with as little content as this and an inefficient, if philosophically attractive, instructional method.

Pre algebra math equations

Polynomial

In mathematics, a polynomial is an expression constructed from variables (also known as indeterminates) and constants, using the operations of addition, subtraction, multiplication, and constant non-negative whole number exponents. For example, x² − 4x + 7 is a polynomial, but x² − 4/x + 7x^3/2 is not, because its second term involves division by the variable x and also because its third term contains an exponent that is not a whole number.

Polynomials are one of the most important concepts in algebra and througho! ut mathematics and science. They are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics, and are used in calculus and numerical analysis to approximate other functions. Polynomials are used to construct polynomial rings, one of the most powerful concep! ts in algebra and algebraic geometry.

Overview

A polynomial is either zero, or can be written as the sum of one or more non-zero terms. The number of terms is finite. These terms consist of a constant (called the coefficient of the term) multiplied by zero or more variables (which are usually represented by letters). Each variable may have an exponent that is a non-negative integer. The exponent on a variable in a term is equal to the degree of that variable in that term. Since x = x¹, the degree of a variable without a written exponent is one. A term with no variables is called a constant term, or just a constant. The degree of a constant term is 0. The coefficient of a term may be any number, including fractions, irrational numbers, negative numbers, and ! complex numbers.

For e! xample,< /p>

is a term. The coefficient is –5, the variables are x and y, the degree of x is two, and the degree of y is one.

The degree of the entire term is the sum of the degrees of each variable in it. In the example above, the degree is 2 + 1 = 3.

A polynomial is a sum of terms. For example, the following is a polynomial:

It consists of three terms: the first is degree two, the second is degree one, and the third is degree zero. Here "− 5x" stands for "+ (−5)x", so the coefficient of the middle term is −5.

When a polynomial in one variable is arranged in the traditional order, the terms of higher degree come before the terms of lower degree. In the first term above, the coefficient is 3, the variable is x, and the exponent is 2. In the second term, the coefficient is –5. T! he third term is a constant. The degree of a non-zero p! olynomia l is the largest degree of any one term. In this example, the polynomial has degree two.

Alternative forms

An expression that can be converted to polynomial form through a sequence of applications of the commutative, associative, and distributive laws is usually considered to be a polynomial. For instance,

is a polynomial because it can be worked out to x³ + 3x² + 3x + 1. Similarly,

is considered a valid term in a polynomial, even though it involves a division, because it is equivalent to and is just a constant. The coefficient of this term is therefore . For similar reasons, if complex coefficients are! allowed, one may have a single term like (2 + 3 i)x³; even though it looks like it should be worked out to two terms, the complex number 2+3i is in fact just a single coefficient in this case that happens to require a "+" to be written down.

Division by an expression containing a variable is not generally allowed in polynomials.^[1] For example,

is not a polynomial because it includes division by a variable. Similarly,

is not a polynomial, because it has a variable exponent.

Since subtraction can be treated as addition of the additive opposite, and since exponentiation to a constant positive whole number power can be treated as repeated multiplication, polynomials can be constructed from constants and variables with just the two operations addition and multiplication.

Polynomial functions

A polynomial function is a function defined by evaluating a polynomial. A function Æ' of one argument is called a polynomial function if it satisfies

for all arguments x, where n is a nonnegative integer and a₀, a₁,a₂, ..., a_n are constant coefficients.

For example, the function Æ', taking real numbers to real numbers, defined by

is a polynomial function of one argument. Polynomial functions of multiple arguments can also be defined, using polynomials in multiple variables, as in

Polynomial functions are an important class of smooth functions.

Polynomial equations

A polynomial equation is an equation in which a polynomial is set equal to another polynomial.

is a polynomial equation. In case of a polynomial equation the variable is considered an unknown, and one seeks to find the possi! ble valu es for which both members of the equation evaluate to the same value (in general more than one solution may exist). A polynomial equation is to be contrasted with a polynomial identity like (x + y)(x – y) = x²–y², where both members represent the same polynomial in different forms, and as a consequence any evaluation of both members will give a valid equality.

Elementary properties of polynomials

1. A sum of polynomials is a polynomial.
2. A product of polynomials is a polynomial
3. The derivative of a polynomial function is a polynomial function
4. Any primitive or antiderivative of a polynomial function is a polynomial function

Polynomials serve to approximate other functions, such as sine, cosine, and exponential.

All polynomials have an expanded form, in which the distributive law has been used to remove all brackets. All polynomials with real or complex coefficients also have a factored form in which the polynomial is written as a product of linear polynomials. For example, the polynomial

is the expanded form of the polynomial

,

which is written in factored form. Note that the constants in the linear polynomial! s (like -3 and +1 in the above example) may be complex numbers in certain cases, even if all coefficients of the expanded form are real numbers. This is because the field of real numbers is not algebraically closed; however, the fundamental theorem of algebra states that the field of complex numbers is algebraically closed.

In school algebra, students learn to move easily from one form to the other (see: factoring).

Every polynomia! l in one variable is equivalent to a polynomial with the form

This form is sometimes taken as the definition of a polynomial in one variable.

Evaluation of a polynomial consists of assigning a number to each variable and carrying out the indicated multiplications and additions. Evaluation is sometimes performed more efficiently using the Horner scheme

In elementary alge! bra, methods are given for solving all first degree and second degree polynomial equations in one variable. In the case of polynomial equations, the variable is often called an unknown. The number of solutions may not exceed the degree, and will equal the degree when multiplicity of solutions and complex number solutions are counted. This fact is called the fundamental theorem of algebra.

A system of polynomial equations is a set of equations in which a given variable must take on the same value everywhere it appears in any of the equations. Systems of equations are usually grouped with a single open brace on the left. In elementary algebra, methods are given for solving a system of linear equations in several unknowns. To get a unique solution, the number of equations should equal the number of unknowns. If there are more unknowns than equations, the system is called underdetermined. If there are more equations than unknowns, the system is called overdetermined. This important subject is studied extensively in the area of mathematics known as linear algebra. Overdetermined systems are common in practical applications. For example, one U.S. mapping survey used computers to solve 2.5 million equations in 40! 0,000 unknowns.^[2]

This article may require cleanup to meet Wikipedia's quality standards. Please improve this article if you can. (Septemb! er 2008)

HISTORY

Determining the roots of polynomials, or "solving algebraic equations", is among the oldest problems in mathematics. However, the elegant and practical notation we use today only developed beginning in the 15th century. Before that, equations are written out in words. For example, an algebra problem from the Chinese Arithmetic in Nine Sections, circa 200 BCE, begins "Three sheafs of good crop, two sheafs of mediocre crop, and one sheaf of bad crop are sold for 29 dou." We would write 3x + 2y + z = 29.

Notation

Main article: History of mathematical notation

The earliest known use of the equal sign is in Robert Recorde's The Whetstone of Witte, 1557. The signs + for addition, − for subtraction, and the use of a letter for an unknown appear in Michael Stifel's Arithemetica integra, 1544. René Descartes, in La géometrie, 1637, int! roduced the concept of the graph of a polynomial equation. He ! populari zed the use of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables, as can be seen above, in the general formula for a polynomial in one variable, where the a 's denote constants and x denotes a variable. Descartes introduced the use of superscripts to denote exponents as well.^[3]

Solving polynomial equations

Every polynomial corresponds to a polynomial function, where f(x) is set equal to the polynomial, and to a polynomial equation, where the polynomial is set equal to zero. The so! lutions to the equation are called the roots of the polynomial and they are the zeroes of the function and the x-intercepts of its graph. If x = a is a root of a polynomial, then (x − a) is a factor of that polynomial.

Some polynomials, such as f(x) = x² + 1, do not have any roots among the real numbers. If, however, the set of allowed candidates is expanded to the complex numbers, every (non-constant) polynomial has at least one distinct root; this follows from the fundamental theorem of algebra.

There is a difference between approximating roots and fin! ding exact roots. Formulas for the roots of polynomials up to ! a degree of 2 have been known since ancient times (see quadratic equation) and up to a degree of 4 since the 16th century (see Gerolamo Cardano, Niccolo Fontana Tartaglia). But formulas for degree 5 eluded researchers. In 1824, Niels Henrik Abel proved the striking result that there can be no general formula (involving only the arithmetical operations and radicals) for the roots of a polynomial of degree 5 or greater in terms of its coefficients (see Abel-Ruffini theorem). This result marked the start of Galois theory which engages in a detailed study of relationships among roots of polynomials.

Numerically solving a polynomial equation in one unknown is easily done on a computer by the Durand-Kerner method or by some other root-finding algorithm. The reduction of equations in several unknowns to equations each in one unknown is discussed in the article on the Buchberger's algorithm. The special case where all the polynomials are of degree one is cal! led a system of linear equations, for which a range of different solution methods exist, including the classical gaussian elimination.

It has been shown by Richard Birkeland and Karl Meyr that the roots of any polynomial may be expressed in terms of multivariate hyperg! eometric functions. Ferdinand von Lindemann and Hiroshi Umemura showed that the roots may also be expressed in terms of Siegel modular functions, generalizations of the theta functions that appear in the theory of elliptic functions. These characterizations of the roots of arbitrary polynomials are generalizations of the methods previously discovered to solve the quin tic equation.

Graphs

A polynomial function in one real variable can be represented by a graph.

• The graph of the zero polynomial

f(x) = 0

is the x-axis.

• The graph of a degree 0 polynomial

f(x) = a₀, where a₀ ≠ 0,

is a horizontal line with y-intercept a₀

• The graph of a degree 1 polynomial (or line! ar function)

f(x) = a₀ + a₁x , where a₁ ≠ 0,

is an oblique line with y-intercept a₀ and slope a₁.

• The graph of a degree 2 polynomial

f(x) = a₀ + a₁x + a₂x², where a₂ ≠ 0

is a parabola.

• The graph of a degree 3 polynomial

f(x) = a₀ + a₁x + a₂! x², + a₃x^{3, wher e a₃ ≠ 0}

is a cubic curve.

• The graph of any polynomial with degree 2 or greater

f(x) = a₀ + a₁x + a₂x² + ... + a_nxⁿ , where a_n ≠ 0 and n ≥ 2

is a continuous non-linear curve.

Polynomial graphs are analyzed in calculus using intercepts, slopes, concavity, and end behavior.

The illustrations below show graphs of polynomials.

Polynomial of degree 2: f(x) = x2 - x - 2 = (x+1)(x-2)	Polynomial of degree 3: f(x) = x3/5 + 4x2/5 - 7x/5 - 2 = 1/5 (x+5)(x+1)(x-2)
Polynomial of degree 4: f(x) = 1/14 (x+4)(x+1)(x-1)(x-3) + 0.5	Polynomial of degree 5: f(x) = 1/20 (x+4)(x+2)(x+1)(x-1)(x-3) + 2

Polynomials and calculus

Main article: Calculus with polynomials

One important aspect of calculus is the project of analyzing complicated functions by means of approximating them with polynomials. The culmination of these efforts is Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial, and the Stone-Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the whole interval as closely as desired by a polynomial. Polynomials are also frequently used to interpolate functions.

Quotients of polynomials are called rational expressions, and functions that evaluate rational expressions are called rational functions. Rational functions are the only functions that can be evaluated on a computer by a fixed sequence of instructions involving operations of addition, multiplication, division, which operations on floating point numbers are usually implemented in hardware. All the other functions that computers need to evaluate, such as trigonometric functions, logarithms and exponential functions, must then be computed in software that may use approximations to those functions on certain intervals by rational functions, and possibly iteration.

Calculating derivatives and integrals of polynomials is particularly simple. For the polynomial

the derivative with respect to x is

and the indefin! ite inte gral is

Abstract algebra

Main article: Polynomial ring

In abstract algebra, one must take care to distinguish between polynomials and polynomial functions. A polynomial f in one variable X over a ring R is defined to be a formal expression of the form

where n is a natural number, the coefficients are elements of R, Here X is a formal symbol, whose powers Xⁱ are at this point just placeholders for the corresponding coefficients a_i, so that the given formal expression is just a way to encode the sequence (a₀,a₁,...), where there is an N such that! a_i=0 for all i>N. Two ! polynomi als sharing the same value of n are considered to be equal if and only if the sequences of their coefficients are equal; furthermore any polynomial is equal to any polynomial with greater value of n obtained from it by adding terms in front whose coefficient is zero. These polynomials can be added by simply adding corresponding coefficients (the rule for extending by terms with zero coefficients can be used to make sure such coefficients exist). Thus each polynomial is actually equal to the sum of the terms used in its formal expression, if such a term a_iXⁱ is interpreted as a polynomial that has zero coefficients at all powers of X other than Xⁱ. Then to define multiplication, it suffices by the distributive law to desc! ribe the product of any two such terms, which is given by the rule

for all elements a, b of the ring R and all natural numbers k and l.

Thus the set of all polynomials with coefficients in the ring R forms itself a ring, the ring of polynomials over R, which is denoted by R[X]. The map from R to R[X] sending r to rX⁰ is an injective homomorphism of rings, by which R is viewed as a subring of R[X]. If R is commu tative, then R[X] is an algebra over R.

One can think of the ring R[X] as arising from R by adding one new element X to R, and extending in a minimal way to a ring in which X satisfies no other relations than the obligatory ones, plus commutation with all elements of R (that is Xr = rX). To do this, one must add all powers of X and their linear combinations as well.

Formation of the polynomial ring, together with forming factor rings by factoring out ideals, are important tools for constructing new rings out of known ones. For instance, the ring (in fact field) of complex numbers! , which can be constructed from the polynomial ring R[X] over the real numbers by factoring out the ideal of multiples of the polynomial X² + 1. Another example is the construction of finite fields, which proceeds similarly, starting out with the field of integers modulo some prime number as the coefficient ring R (see modular arithmetic).

If R is commutative, then one can associate to every polynomial P in R[X], a polynomial function f wi! th domain and range equal to R (mo re generally one can take domain and range to be the same unital associative algebra over R). One obtains the value f(r) by everywhere replacing the symbol X in P by r. One reason that algebraists distinguish between polynomials and polynomial functions is that over some rings different polynomials may give rise to the same polynomial function (see Fermat's little theorem for an example where R is the integers modulo p). This is not the case when R is the real or complex numbers and therefore many analysts often don't separate the two concepts. An even more important reason to distinguish between polynom! ials and polynomial functions is that many operations on polynomials (like Euclidean division) require looking at what a polynomial is composed of as an expression rather than evaluating it at some constant value for X. And it should be noted that if R is not commutative, there is no (well behaved) notion of polynomial function at all.

Divisibility

In commutative algebra, one major focus of study is divisibility among polynomials. If R is an integral domain and f and g are polynomials in R[X], it is said that f divides g if there exists a polynomial q in R[X] such that f q = g. One can then show that "every zero gives rise to a linear factor", or more formally: if f is a polynomial in R[X] and r is an element of R such that f(r) = 0, then the polynomial (X − r) divides f. The converse is also true. The quotient can be computed using the Horner scheme.

If F is a field and f and g are polynomials in F[X] with g ≠ 0, then there exist unique polynomials q and r in F[X] with

and such that the degree of r is smaller than the degree of g. The polynomials q and r are uniquely determined by f and g. This is called "division with remainder" or "polynomial long division" and shows that the ring F[X] is a Euclidean domain.

Analogously, polynomial "primes" (more correctly, irreducible polynomials) can be defined which cannot be factorized into the product of two polynomials of lesser degree. It is not easy to determine if a given polynomial is irreducible. One can start by simply checking if the polynomial has linear factors. Then, one can check divisibility by some other irreducible polynomials. Eisenstein's criterion can also be used in some cases to determine irreducibility.

See also: Greatest common divisor of two polynomials.

Classifications

The most important classification of polynomials is based on the number of distinct variables. A polynomial in! one variable is called a univariate polynomial, a polynomial in more than one variable is called a multivariate polynomial. These notions refer more to the kind of polynomials one is generally working with than to individual polynomials; for instance when working with univariate polynomials one does not exclude constant polynomials (which may result for instance from the subtraction of non-constant polynomials), although strictly speaking constant polynomials do not contain any variables at all. It is possible to further classify multivariate polynomials as bivariate, trivariate etc., according to the number of variables, but this is rarely done; it is more common for instance to say simply "polynomials in x, y, and z". A (usually multivariate) polynomial is called homogeneous of degree n if all! its terms have degree ! n .

Univariate polynomials have many properties not shared by multivariate polynomials. For instance, the terms of a univariate polynomial are completely ordered by their degree, and it is conventional to always write them in order of decreasing degree. A univariate polynomial in x of degree n then takes the general form

where c_n, c_n-1, ..., c₂, c₁ and c₀ are constants, the coefficients of this polynomial. Here the term c_nxⁿ is called the leading term and its coefficient c_n the leading coefficient; i! f the leading coefficient is 1, the univariate polynomial is called monic. Note that apart from the leading coefficient c_n (which must be non-zero or else the polynomial would not be of degree n) this general form allows for coefficients to be zero; when this happens the corresponding term is zero and may be removed from the sum without changing the polynomial. It is nevertheless common to refer to c_i as the coefficient of xⁱ, even when c_i happens to be 0, so that xⁱ does not really occur in any term; for instance one can speak of the constant term of the polynomial, meaning c₀ even if it should be zero.

Polynomials can similarly be class! ified by the kind of constant values allowed as coefficients. ! One can work with polynomials with integral, rational, real or complex coefficients, and in abstract algebra polynomials with many other types of coefficients can be defined. Like for the previous classification, this is about the coefficients one is generally working with; for instance when working with polynomials with complex coefficients one includes polynomials whose coefficients happen to all be real, even though such polynomials can also be considered to be a polynomials with real coefficients.

Polynomials can further be classified by their degree and/or the number of non-zero terms they contain.

Polynomials classified by degree

Degree	Name	Example
−∞	zero	0
0	(non-zero) const! ant	1
1	linear	x + 1
2	quadratic	x2 + 1
3	cubic	x3 + 1
4	quartic (or biquadratic)	x4 + 1
5	quintic	x5 + 1
6	sextic or hexic	x6 + 1
7	septic or heptic	x7 + 1
8	octic	x8 + 1
9	nonic	x9 + 1
10	decic	x10 + 1

Usually, a polynomial of degree 4 or higher is referred to as a polynomial of degree n, although the phrases quartic polynomial and quin! tic polynomial are also used. The names for degrees higher than 5 are even less common. The names for the degrees may be applied to the polynomial or to its terms. For example, a constant may refer to a zero degree polynomial or to a zero degree term.

The polynomial 0, which may be considered to have no terms at all, is called the zero polynomial. Unlike other constant polynomials, its degree is not zero. Rather the degree of the zero polynomial is either left explicitly undefined, or defined to be negative (either –1 or –∞).^[4] The latter convention is important when defining Euclidean division of polynomials.

Polynomials classified ! by numbe r of non-zero terms

Number of non-zero terms	Name	Example
0	zero polynomial	0
1	monomial	x2
2	binomial	x2 + 1
3	trinomial	x2 + x + 1

The word monomial can be ambiguous, as it is also often used to denote just a power of the variable, or in the multivariate case product of such powers, without any coefficient. Two or more terms which involve the same monomial in the latter sense, in other words which differ only in the value of th! eir coefficients, are called similar terms; they can be combined into a single term by adding their coefficients; if the resulting term has coefficient zero, it may be removed altogether. The above classification according to the number of terms assumes that similar terms have been combined first.

Polynomials associated to other objects

Polynomials are frequently used to encode information about some other object. The characteristic polynomial of a matrix or linear operator contains information about the operator's eigenvalues. The minimal polynomial of an algebraic element records the simplest algebraic relation satisfied by that element. The chromatic polynomial of a graph counts the number of proper colourings of that graph.

Extensions of the concept of a polynomial

Polynomials can involve more than one variable, in which they are called multivariate. Rings of polynomials in a finite nu! mber of variables are of fundamental importance in algebraic geometry which studies the simultaneous zero sets of several such multivariate polynomials. These rings can alternatively be constructed by repeating the construction of univariate polynomials with as ceofficient ring another ring of polynomials: thus the ring R[X,Y] of polynomials in X and Y can be viewed as the ring (R[X])[Y] of polynomials in Y with as coeffcients polynomials in X, or as the ring (R[Y])[X] of polynomials in X with as coeffcients polynomials in Y. These identifications are compatible with arithmetic operations (they are isomorphisms of rings), but some notions such as degree or whether a polynomial is considered monic can change be! tween these points of view. One can construct rings of polynom! ials in infinitely many variables, but since polynomials are (finite) expressions, any individual polynomial can only contain finitely many variables.

Laurent polynomials are like polynomials, but allow negative powers of the variable(s) to occur.

Rational functions are formal quotients of polynomials (they are formed from polynomials just as rational numbers are formed from integers, writing a fraction of two of them; fractions related by the cancelling of common factors are identified with each other). The rational functions contain the Laurent polynomials, but do not limit denominators to ! be powers of a variable.

Formal power series are like polynomials, but allow infinitely many nonzero terms to occur, so that they do not have finite degree. Unlike polynomials they cannot in general be explicitly and fully written down (just like real numbers cannot), but the rules for manipulating their terms are the same as for polynomials.

How to factor complex polynomials