Chapter 1

A polynomial is a mathematical expression, whose terms are the products of numerical coefficients and variable bases with nonnegative exponents, and separated by plus or minus signs.

Below is an example of a term.

$latex 8x^{2} $
The coefficient is 8, the base is x, and the exponent (or power) is 2.
The term is read, “eight x to the second power, or eight x squared.”

Below are examples of polynomials in one variable (only one letter of the alphabet is used),

degree 0 (constant polynomial):

$latex -6 $

degree 1 polynomial (or linear):

$latex 5x – 6 $

degree 2 polynomial (or quadratic):

$latex 3x^{2} + 5x – 6 $

degree 3 polynomial (or cubic):

$latex -2x^{3} + 3x^{2} + 5x – 6 $

degree 4 polynomial (or quartic):

$latex x^{4} – 2x^{3} + 3x^{2} + 5x – 6 $

degree 5 polynomial (or quintic):

$latex 7x^{5} + x^{4} – 2x^{3} + 3x^{2} + 5x – 6 $

The degree of a polynomial, is the highest exponent (power) in the expression.Below are more examples of polynomials in one variable. Remember, the degree is the highest exponent, not the number of terms. Also, since $latex x^{1} $ is so common, it is usually written as just x.

degree 1: $latex 5x $ $latex x-9 $ $latex -7x+4x $
degree 2: $latex 8x^{2} $ $latex x^{2}-4 $ $latex -4x^{2}+x+3 $
degree 3: $latex x^{3} $ $latex 6x^{3}-27x $ $latex 7x^{3}+5x^{2}-x+1 $

The following examples are not polynomials.

$latex 5x^{3}+2x^{-4} $ $latex 3x^{2}+x^{0.5} $ $latex 4x^{2}-\frac{3}{x} $
  • Polynomials can have rational ( 2/3 ), radical ( √2 ), transcendental ( π ), or complex ( 4 – 5i) coefficients as well.
  • Polynomials can even have polynomial ( 3n + 6 ) coefficients.
  • Polynomials can also have more than one variable ( x2 + xy – y2 or 8xy2z5 for instance).

For now though, we’re going to stick with integer ( . . . -3, -2, -1, 0, 1, 2, 3, . . . ) coefficients, and one variable.  All polynomials are functions. A function is a rule where any input value gives a unique output value.
In the polynomial 3x – 1 for instance, no matter what value you input for x, you will only get one value for the output.

input

polynomial rule 3x – 1 one and only output

$latex 7 $

$latex 3( 7 ) – 1 $

$latex 20 $

$latex \frac{2}{3} $

$latex 3\left( \frac{2}{3} \right)-1 $

$latex 1 $

$latex -0.8 $ $latex 3( -0.8 ) – 1 $

$latex 3.4 $

The notation f( x ), which is read “f of x”, is called function notation. It’s an easy way to give the rule and the input value.
For example, f( x ) = x2 + 4 , means the variable being used is x, the polynomial rule x2 + 4 is a function, and its name is f.
The notation f( 3 ), which is read “f of 3”, is a way of asking what the output would be, if we plugged in a 3 for x, in rule f.

Given $latex f(x ) = x^{2} + 4, f( 3 ) = ( 3 )2 + 4 = 9 + 4 = 13 $

This computer shaped icon indicates the directions which follow, are part of this 8th grade course. Learners are expected to fulfill the requirements of those directions, and to sustain a determined effort over a period of time when necessary. Run program 101 until you can correctly determine at least 5 of the 6 output values.

A book shaped icon that indicates the following directions are opportunities for further information, exploration, or learning. These opportunities are optional, unless the learner finds them helpful, interesting, and/or necessary for algebra credit. For additional information and a couple of examples, open the pdf “functionnotation”.

The pencil shaped icon indicates the following directions involve recording and defending work and discoveries on paper, in order to fulfill the requirements of the previous directions (templates for doing this will be provided by the teacher). Use the pdf “programnotes” to record your work and any notes. Make sure to neatly copy down the original problems, show any work involved, and write your answers down, before typing your answers into the program.

The smiley face icon is a friendly stop sign, to remind the learner to defend their work, participate in mathematical discourse with at least one other student, keep the applet/program open, and then check with their math teacher before continuing After correctly determining at least 5 of the 6 output values, and neatly recording your work, go on to the next page.
For polynomials in one variable, there are relationships or patterns between the input values and the resulting output values. Knowing what these relationships or patterns are, will make polynomials easier to understand. Polynomials in one variable have an infinite number of possible input values, with an infinite number of resulting output values, however. It could be very time consuming to choose and calculate enough input and output values, before noticing the relationships or patterns between them. A quicker way to see the relationships or patterns, is to graph the input and output values.an example of an ordered pair.

An input value followed by its resulting output value, is an example of an ordered pair.

Looking at the earlier example f( x )  =  x2 + 4, f( 3 )  =  ( 3 )2 + 4  =  9 + 4  =  13, gives us the ordered pair (3, 13). To graph the ordered pair (3, 13), we would simply plot a point at those coordinates (3 units to the right, and 13 units up from the origin).

computer icon Run program 102 until you can correctly give the coordinates for at least 18 of the 20 points shown.
computer icon Run program 103 until you can correctly graph or plot the points for at least 8 of the 10 ordered pairs.
pencil icon Use a copy of the pdf “programnotes2g” to plot and label the axes, and the 10 points given in program 103.
book icon For more information, open the pdf “coordinates”.

smiley face icon After meeting the 3 requirements above, go on to the next page.

If we plot enough ordered pairs, we end up graphing the polynomial, and displaying the input/output patterns. Below are snapshots of a program, which will allow you to graph polynomials.

snapshots of a program, which will allow you to graph polynomials.

computer icon Run program 104 until you feel comfortable using it to graph polynomials.

smiley face icon After you feel comfortable using this program to graph polynomials, go on to the next page.

Below are snapshots of an applet, which will also allow you to graph polynomials.

snapshots of an applet, which will also allow you to graph polynomials.

computer icon Run the applet “tracer.html” until you feel comfortable using it to graph polynomials.

The smiley face icon is a friendly stop sign, to remind the learner to defend their work, participate in mathematical discourse with at least one other student, keep the applet/program open, and then check with their math teacher before continuing After you feel comfortable using this applet to graph polynomials, go on to the next page.

 

To the right and below are snapshots of another application, which will allow you to graph polynomials. snapshots of another application, which will allow you to graph polynomials.

computer icon Run the Grapher program “polynomials.gcx” until you feel comfortable using it to graph polynomials.

smiley face icon After you comfortable using this application to graph polyonomials, go on to the next page.

Now that you feel comfortable using at least one of the following three programs,
we can now begin exploring polynomials.

icons depicting three graphing applications

program 104                         applet “tracer.html”             grapher “polynomials.gcx”

computer icon Run one or more of the above three graphing applications,
to explore and defend an answer, to at least 8 of the 17 questions found in the pdf “graphingpolynomials” .

book icon Use a copy of the pdf “booknotes” to record and defend your answers and discoveries.
smiley face icon After recording and defending an answer, to at least 8 of the 17 questions, go on to the next page.Polynomials will be the common theme throughout this book. This will allow for a deeper understanding of the objects and ideas presented, their connections, and mathematical processes in general.

Now that you are more familiar with polynomials, a beneficial definition of (and approach to) algebra can be shared.

Algebra– the logical and creative endeavor of translating, transforming, and discovering polynomial patterns.

Below are further explanations to better understand the above definition.

  • Translating – converting words and/or ideas into polynomial expressions or equations, and then back again.
  • Transforming – changing polynomial expressions or equations into more useful forms without losing their identities.
  • Discovering – exploring new polynomial patterns and/or connections between mathematical objects.
  • Logical – the path to discovery is made with mathematically sound steps which others can follow.
  • Creative – the path to discovery is not always straight, and may require surprising approaches from different directions. Endeavor – the path to discovery requires sustaining a determined effort over a period of time.

The following examples will give you a peek at the versatility of polynomials.That any constant or variable can be treated as a polynomial as well, will allow for geometric interpretations and a better understanding of the mathematical objects and ideas introduced in this book. This will be especially obvious for solving equations and systems of equations in later chapters. Probability is another area where this dual nature of expressions will be helpful.

If a basketball player is fouled in the act of shooting, and the ball goes in anyway, that player is awarded one free throw. If the team that committed the foul has 7 or more fouls though, the fouled player is awarded a one and one free throw attempt. What this means is, if the player makes the first free throw, that player gets to take another free throw. If the fouled player misses the first free throw however, that player does not get to take another free throw.In other words, the fouled player can end up with 2 points (if both free throws go in), 1 point (if only the first free throw goes in), or 0 points (if the first free throw is missed).

If a fouled player is awarded a one and one, and is a 60% free throw shooter,how many points would you expect that player to get, 2, 1, or 0?

 

1 in 1 freethrows iconbook icon After choosing 2, 1, or 0, run the applet “1n1freethrows.html” to see how treating the probabilities as polynomials, helps to explain the answer.

Any ratio can be thought of as the coefficient to a degree one polynomial. This means percent problems, rate problems, and conversion problems, all have geometric representations which can lead to a better understanding of these ideas.

image of graph and question what is 20% of 85?

What is 20% of 85?

 

book icon To see how using 20% as the coefficient of a linear polynomial can show the answer to the above question, run the applet “20percent.html” .
On an even more basic level, the digits of every positive integer number, are actually the coefficients of polynomials. Knowing this will help us to understand our number system and many of arithmetic’s algorithms. In other words, studying polynomials will allow us to understand more, and memorize less.
polynomial

book icon To find out how to solve the following polynomial equation for x, run the applet “numbersystems.html”.In business, polynomials are used to represent cost and revenue, or supply and demand, which then can be used to find break even points or equilibrium prices.In engineering, polynomials are used to describe moments of inertia, and estimating the safe working load of ropes.In the medical field, polynomials are used to calculate medication dosages.Polynomials were used as part of the code for most of the programs and applets in this book. In any field where things are being quantified, there are formulas that are either polynomial, contain polynomials, or can be better understood from being familiar with polynomials.If you open a CD (certificate of deposit) at a bank, a polynomial will be able to tell you how much money you will have after it matures (after a set amount of time has passed).

book icon To see what happens to a deposit of $1,000 after 5 years, run the applet “CDs.html”.Geometric formulas for perimeter, area , and volume are often just polynomial equations ( Pcircle = 2 π r, Acircle = π r2, and Vsphere = 4/3 π r3 for instance ).Understanding polynomials will give us more insight into what these formulas actually represent. The famous Pythagorean Theorem $latex A^{2} + B^{2} = C^{2} $ , for instance, is just a degree 2 polynomial equation. With 3 variables, the graph of the Pythagorean Theorem can be shown in 3-D, where the A and B are represented by the twohorizontal X and Y axes, and C is represented by the vertical axis

Choosing a particular value for C, is like taking a cross section (cutting the cone with a horizontal plane). As It turns out, the bird’s eye view of the 3-D model, is actually the circular 2-D model of the Pythagorean Theorem. By exploring both the 3-D and the 2-D models, we can see the Pythagorean Theorem is related to circles.

book icon To better see this relationship, run the applet “pythagcircle.html”.

Physics formulas are very often polynomial equations as well.Being familiar with polynomials, will allow you to better understand these formulas and how the world works. For instance, because there is a quadratic relationship between time and distance for objects in free fall, the trajectories (paths) of objects being thrown, launched, or dropped near the surface of the earth, can be modeled with degree 2 polynomials.

book icon To model the trajectory (path) of the water coming out of a school bubbler with a quadratic (degree 2 polynomial), run the applet “fountain.html”old school water fountain.
By iterating (inputting a value to get an output value, then using that output value as the next input value, and repeating) polynomial equations, fractal geometry, dynamical systems, and the natural world can be better understood. The Mandelbrot (MAN-del-brot) set, named after Benoit Mandelbrot (ben-WAH MAN-del-bro), and the related Julia sets, named after Gaston Julia (ga-STONE JOO-lia), are amazingly intricate and beautiful mathematical objects, which we are able to estimate due to the speed of computers.

 

Chapter 3 icon

Filled in Julia set, whose attractors are the solutions to the iterated polynomial equation (zn+11)2 + (.32 + .043i) = zn, discovered by H. O. Peitgen and P. H. Richter

book icon To create the Julia set seen on the right, which was discovered by H. O. Peitgen and P. H. Richter, run program 105 .It is made by iterating the degree 2 polynomial z2 + c for a particular c value and a range of z values, where z and c are complex numbers.

 

book icon To learn about complex numbers, open the pdf “complex numbers”.

 

Michael Barnsley’s fernA book shaped icon that indicates the following directions are opportunities for further information, exploration, or learning. These opportunities are optional, unless the learner finds them helpful, interesting, and/or necessary for algebra credit. To create Michael Barnsley’s (MY-kull BARNS-lee’s) fern, seen on the right, run program 110 .

 

It is made by iterating four things called 2-D affine transformations, which look like degree one polynomials when written in the form Ax + t.

 

The A represents something called a 2 by 2 matrix and t represents something called a 2 by 1 matrix, where these matrices (plural for matrix) describe rotations, reflections, magnifications, and translations.

 

In the 800’s, a book with six chapters on solving degree 1 and degree 2 polynomial equations led to the modern word “algebra”.In the 1500’s, work on solving degree 3 and degree 4 polynomial equations led to complex numbers. In the 1800’s, it was shown there were no formulas for solving degree 5 or higher polynomial equations, and modern algebra (a branch of higher level mathematics) was born. The Fundamental Theorem of Algebra has to do with the existence of roots (solutions) of polynomial equations. An algebraic number is a number which is the root of a polynomial equation with integer coefficients. Polynomials have provided a wealth of research in mathematics and continue to do so to this day. One of the reasons for this, is their ability to model other functions and patterns in general.  For instance, polynomials can produce valid answers for any of the Mensa (the high I.Q. group) problems of the following type.

 

What comes next?
1, 2, 5, 10, 17, ____
2, 4, 12, 26, 46, ____
3, -7, 8, 8, 10, ____

book icon To try solving more of these puzzles, run program 111 .

There is quite a surprising twist to the answers to the above three puzzles, and puzzles like them. By the end of the book, we will be able to figure out these answers and discover this surprise. At this point however, much more exploration of polynomials is needed. We will start by finding out what happens when we add, subtract, or multiply a polynomial,
by another polynomial.

Below are snapshots of two applets, which will allow you to explore adding and subtracting polynomials.

snapshots of two applets, which will allow you to explore adding and subtracting polynomials.

computer icon Run the applets “addpolys.html” and “subpolys.html” to explore and answer the 5 questions in the pdf “addsubpolys”.

pencil icon Open the pdf “addsubpolys” to record and defend your answers to the 5 questions given.
smiley face icon After recording and defending your answers to the 5 questions, go on to the next page.

Below are snapshots of a program, which will randomly generate 10 polynomial subtraction problems.

snapshots of a program, which will randomly generate 10 polynomial subtraction problems.

computer icon Using what you have learned from answering the 5 questions in the pdf “addsubpolys”, run program 112 and program 113 , until you correctly add or subtract at least 8 of the 10 problems from each program.
pencil icon Use a copy of the pdf “programnotes” to record the original problems, all of your work, and any notes.Make sure to neatly copy down the original problems, show any work involved, and write your answers down before typing your answers into the program.The reason for taking your time and showing all of your work is, knowing how to get correct answers, is not good enough. Your goal is to gain a deep understanding of the ideas and be able to make connections to other ideas. This is made easier by getting the needed practice of writing and thinking in the language of mathematics.After entering your answers, the programs will give you immediate feedback.

How many problems you get right the first time around, is not as important as, what you do with that information.

smiley face icon After sustaining a determined effort over a long period of time, and correctly answering at least 8 out of 10 problems from each program, go on to the next page.

Below is a snapshot of an applet, which will allow you to explore multiplying polynomials.

snapshot of an applet, which will allow you to explore multiplying polynomials.

computer icon Run the applet “multpolys.html” to explore and answer the 5 questions in the pdf “multpolys”.
pencil icon Use a copy of the pdf “multpolys” to record and defend your answers to the 5 questions given.
smiley face icon After sustaining a determined effort over a long period of time, and defending your answers to the 5 questions, go on to the next page.

Below are snapshots of 8 programs, which will randomly generate polynomial multiplication problems.

snapshots of 8 programs, which will randomly generate polynomial multiplication problems.

computer icon Using what you have learned from exploring the 5 questions in the pdf “multpolys”, run program 114 through program 121 , starting with program 114 , with 2 or fewer oopsy daisies per program.
book icon For more information and examples, open the pdf “distributive“.
pencil icon Use a copy of the pdf “programnotes” to record the original problems, all of your work, and any notes.Make sure to neatly copy down the original problems, show any work involved, and write your answers down before typing your answers into the program.

smiley face icon After sustaining a determined effort over a long period of time, and being able to finish all 8 programs, with 2 or fewer oopsy-daisies per program, go on to the next page.Earlier in the chapter, we were introduced to function notation. We also had the opportunity to evaluate polynomial expressions with numerical input values. It turns out though, inputs and outputs don’t have to be numbers, they can be polynomial expressions themselves as well.As we discovered, adding or subtracting polynomials is just a matter of collecting like terms, and multiplying polynomials is just an example of the distributive property.

Below are snapshots of a program, which will randomly generate 10 challenges, written using function notation.

snapshots of a program, which will randomly generate 10 challenges, written using function notation.

computer icon Run program 122 to see how many of the 10 challenges you can successfully complete.
pencil icon Use a copy of the pdf “programnotes” to record the original functions, all of your work, and any notes.

book icon For more information and examples with the work shown, open the pdf “functionchallenge”.

smiley face icon After sustaining a determined effort over a long period of time, go on to the next page.

Being able to simplify polynomial expressions correctly, will be an important part of this course.

Since exponents are used to represent the repeated multiplication of variables in polynomial expressions, we discovered when multiplying terms with like bases, we can simplify by adding the exponents

$latex x^{2 }x^{3} $ = $latex (x*x)(x*x*x) $ = $latex x*x*x*x*x = x^{5} $

This is the same reason we can simplify terms being raised to a power, by multiplying the exponents.

$latex (x2)^{3} $ = $latex (x^{2})(x^{2})(x^{2}) = (x*x)(x*x)(x*x) = x*x*x*x*x*x = x^{6} $

Below are two examples, where the monomials (the terms of polynomials, or polynomials with only one term) have two different bases.

$latex 5x^{2}y^{3}(4x^{8}y^{4}) $ = $latex 20x^{(2+8)}y^{(3+4)} $ = $latex 20x^{10}y^{7} $

$latex (2a^{3}b^{5})^{3} $ = $latex 8a^{(3*3)}b^{(5*3)} $ = $latex 8a^{9}b^{15} $

Below are a couple of examples of typical math contest problems, which can be solved quickly without calculators.

examples of typical math contest problems, which can be solved quickly without calculators.

Since division undoes multiplication, when dividing monomials with like bases, we can simplify by subtracting the exponents.

Example of dividing monomials with like bases, we can simplify by subtracting the exponents

Below are a couple of examples, where the monomials have two or more different bases.

Examples of monomials with two or more different bases

Simplifying by subtracting the exponents, means we can end up having an exponent which is zero or negative.

Example of exponent which is zero or negative

At first, this might sound a little funny.An exponent of zero though, just means all of the bases were reduced to a value of 1(with zero bases left over).

A negative exponent, just means all of the bases left over after reducing, are below the fraction bar. In other words, negative exponents represent reciprocals (not negatives).

Below is an example, where the monomials have three different bases.

Example where the monomials have three different bases.
By writing everything out the long way, we can see subtracting the exponents, is just a quicker way to simplify by reducing (after reducing, we end up with 2 a’s left over on the top, 0 b’s left over, and 6 c’s left over in the bottom).

Example of writing things out the long way

Although not monomials, writing expressions with negative exponents, allows us to write fractions without fraction bars.

Example of writing expressions with negative exponents, allows us to write fractions without fraction bars.

Understanding negative exponents represent reciprocals, also helps us to better understand scientific notation for small numbers.

image of numbers depicting the Understanding negative exponents represent reciprocals, also helps us to better understand scientific notation for small numbers.

Below is a snapshot of a program, which will randomly generate 18 expressions to be simplified.

snapshot of a program, which will randomly generate 18 expressions to be simplified.

computer icon Run program 123 until you can simplify at least 16 out of the 18 expressions.

pencil icon Use a copy of the pdf “programnotes” to neatly record the expressions, before and after simplifying.
smiley face icon After correctly simplifying at least 16 of the 18 expressions, and neatly recording them along with the resulting monomials, go on to the next page.

book icon If the picture at the beginning of the chapter looked interesting, run one or more of the following programs to explore cellular automata, based on John Conway’s game of Life.

    • program 128
    • program 129
    • program 130

Below is a snapshot of the chapter 1 programs menu program.

snapshot of the chapter 1 programs menu program.

computer icon Run the program “1menu.bas” , or its alias to access every program used in chapter 1.

 

“Mathematics seems to endow one with something like a new sense” — Charles Darwin —

“Simply put, aesthetic and intellectual fulfillment requires that you know about mathematics.” — Jerry P. King —

“One of the most important concepts in all of mathematics is that of function.” — T.P. Dick and C.M. Patton —

“Algebra is generous; she often gives more than is asked of her.” — D’Alembert —

“Algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world.” — Alfred North Whitehead —

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