Operations with Polynomial Expressions and Introduction to Graphing Polynomials

In this lesson we will review mathematical techniques for working with polynomials.

We will review vocabulary pertaining to polynomials functions. We will also make some important connections between the degree of the polynomial and its leading coefficient and the behavior of the graph of the polynomial.

What is a Polynomial?

We will start our review with polynomials in one variable.

Polynomials in one variable are algebraic expressions that consist of terms in the form

LaTeX: ax^n

 where n is a non-negative (i.e. positive or zero) integer and 𝑎 is a real number and is called the coefficient of the term. The degree of a polynomial in one variable is the largest exponent in the polynomial.

Note that we will often drop the “in one variable” part and just say polynomial.

Here are examples of polynomials and their degrees.

$LaTeX: 5x^{12}-2x^6+x^5-198x+1$	degree: 12
	degree: 4
$LaTeX: 56x^{23}$	degree: 23
	degree: 1
	degree: 0

So, a polynomial doesn’t have to contain all powers of 𝑥 as we see in the first example. Also, polynomials can consist of a single term as we see in the third and fifth example.

We should probably discuss the final example a little more. This really is a polynomial even it may not look like one. Remember that a polynomial is any algebraic expression that consists of terms in the form

LaTeX: ax^n

Another way to write the last example is

$- 8 x^{0}$

Written in this way makes it clear that the exponent on the x is a zero (this also explains the degree…) and so we can see that it really is a polynomial in one variable.

Here are some examples of things that aren’t polynomials.

$LaTeX: 4x^6+15x^{-8}+1$

$LaTeX: 5\sqrt{x}-x+x^2$

$LaTeX: \frac{2}{x}+x^3-2$

Algebraic Operations with Polynomial Expressions

Recall that we can add and subtract polynomial expressions. Here are two examples.

Worked Example
Self Check
Self Check

Worked Example:

Perform the indicated operation for each of the following.

(A) Add LaTeX: 6x^5-10x^2+x-45 to LaTeX: 13x^2-9x+4

(B) Subtract from LaTeX: 5x^2-9x^2+x-3 from LaTeX: x^2+x+1

Example A:

The first thing that we should do is actually write down the operation that we are being asked to do.

$LaTeX: \left(6x^5-10x^2+x-45\right)+\left(13x^2-9x+4\right)$

In this case the parenthesis are not required since we are adding the two polynomials. They are there simply to make clear the operation that we are performing. To add two polynomials all that we do is combine like terms. This means that for each term with the same exponent we will add or subtract the coefficient of that term.

In this case this is,

$LaTeX: \left(6x^5-10x^2+x-45\right)+\left(13x^2-9x+4\right)=6x^5+\left(-10+13\right)x^2+\left(1-9\right)x-45+4$

$LaTeX: 6x^5+\left(-10+13\right)x^2+\left(1-9\right)x-45+4=6x^5+3x^2-8x-41$

Example B:

Again, lets write down the operation we are doing here. We will also need to be very careful with the order that we write things down in. Here is the operation

$LaTeX: x^2+x+1-\left(5x^3-9x^2+x-3\right)$

This time the parentheses around the second term are absolutely required. We are subtracting the whole polynomial and the parenthesis must be there to make sure we are in fact subtracting the whole polynomial.

In doing the subtraction the first thing that we'll do is distribute the minus sign through the parenthesis. This means that we will change the sign on every term in the second polynomial. Note that all we are really doing here is multiplying a "-1" through the second polynomial using the distributive law. After distributing the minus through the parenthesis we again combine like terms.

Here is the work for this problem.

$LaTeX: x^2+x+1-\left(5x^3-9x^2+x-3\right)=x^2+x+1-5x^3+9x^2-x+3$

LaTeX: x^2+x+1-5x^3+9x^2-x+3=-5x^3+10x^2+4

Note that sometimes a term will completely drop out after combing like terms as the LaTeX: x did here. This will happen on occasion so don't get excited about it when it does happen.

INSTRUCTIONS: Perform the indicated operation and identify the degree of the results.

Add LaTeX: 4x^3-2x^2+1 to LaTeX: 7x^2+12x

Solution

Here is the operation we're being asked to perform.

$LaTeX: \left(4x^3-2x^2+1\right)+\left(7x^2+12x\right)$

Note that the parenthesis are only there to illustrate how each polynomial being used in the operation and are not needed (or used) in general.

Here's the result of the operation.

$LaTeX: \left(4x^3-2x^2+1\right)+\left(7x^2+12x_{}\right)=4x^3+5x^2+12x+1$

Remember the degree of a polynomial is just the largest exponent in the polynomial and so the degree of the result of this operation is three.

INSTRUCTIONS: Perform the indicated operation and identify the degree of the result.

Subtract LaTeX: 4z^6-3z^2+2z from LaTeX: -10z^6+7z^2-8

Solution

Here is the operation we're being asked to perform.

$LaTeX: -10z^6+7z^2-8-\left(4z^6-3z^2+2z\right)$

Be careful with the order here! We are subtracting the first polynomial from the second and that implies the order we've got here. Also be careful with the parenthesis on the second polynomial. We are subtracting the whole polynomial and so we need to have the parenthesis to do that.

Here's the result of the operation.

$LaTeX: -10z^6+7z^2-8-\left(4z^6-3z^2+2z\right)=-10z^6+7z^2-8-4z^6+3z^2-2z$

LaTeX: -10z^6+7z^2-8-4z^6+3z^2-2z=-14z^6+10z^2-2z-8

Remember the degree of a polynomial is just the largest exponent in the polynomial and so the degree of the result of this operation is six.

Multiplying Polynomials

Recall from your previous algebra class that we can multiply polynomials. Here are some examples:

Worked Example
Self Check

Worked Example:

(A) $LaTeX: \left(3x+5\right)\left(3x-5\right)$

(B) $LaTeX: \left(3x+1\right)\left(2-9x^2\right)$

Example A:

We can use FOIL on this one so let's do that.

$LaTeX: \left(3x+5\right)\left(3x-5\right)=9x^2-15x+15x-25$

LaTeX: 9x^2-15x+15x-25=9x^2-25

In this case the middle terms drop out.

Example B:

All we need to do is FOIL out the two polynomials. Here is the result of doing that.

$LaTeX: \left(3x+1\right)\left(2-9x^2\right)=2+6x-9x^2-27x^3$

INSTRUCTIONS: Perform the indicated operation and identify the degree of the results.

$LaTeX: \left(w^2+2\right)\left(3w^2+w\right)$

Solution

All we need to do is FOIL out the two polynomials. Here is the result of doing that.

$LaTeX: \left(w^2+2\right)\left(3w^2+w\right)=3w^4+w^3+6w^2+2w$

Remember the degree of a polynomial is just the largest exponent in the polynomial and so the degree of the result of this operation is four.

Factoring Polynomials

Recall that we worked on factoring quadratic polynomials in Module 2 of this course. Here is an example to refresh your memory of that work.

Factoring will be important to us as learn more about polynomials and their graphs.

Worked Example
Self Check

Worked Example:

Factor the following polynomial.

LaTeX: 5x^2+14x-3

Soultion:

There are only two positive factors of 5 so the initial form for the factoring will be,

$LaTeX: \left(5x+_-\right)\left(x+_-\right)$

and the factors of -3 are,

$LaTeX: \left(-1\right)\left(3\right)$ $LaTeX: \left(1\right)\left(-3\right)$

After some trial and error we see that the correct factoring will then be,

$LaTeX: 5x^2+14x-3=\left(5x-1\right)\left(x+3\right)$

Self Check:

LaTeX: z^2-10z+21

Solution

The initial form for the factoring will be,

$LaTeX: \left(z+_-\right)\left(z+_-\right)$

and the factors of 21 are,

$LaTeX: \left(-1\right)\left(21\right)$ $LaTeX: \left(1\right)\left(21\right)$ $LaTeX: \left(-3\right)\left(-7\right)$ $LaTeX: \left(3\right)\left(7\right)$

Now, recalling that we need the pair of factors from the above list that will add to get -10. So, we can see that the correct factoring will then be,

$LaTeX: z^2-10z+21=\left(z-3\right)\left(z-7\right)$

Instructional Content:

Part I. Using Leading Coefficients and The Degree of a Polynomials to Predict the End Behavior of Graphs

There is an important connection between the degree of a polynomial and the behavior of the graph. This connection helps us identify what is known as the end behavior of the polynomial function. Consider the graphs shown below.

Example
Self Check
Self Check

Each of the polynomials show in the graphs have an even degree. We can see that if the degree of the polynomial is even and the leading coefficient is positive, then the graph of the function extends up on both ends as in graphs A, B and C.

Graph A:

$LaTeX: f\left(x\right)=x^4-3x^2+2$

Degree: 4

Leading Coefficient: 1

Graph B:

$LaTeX: f\left(x\right)=x^6-x^3$

Degree: 6

Leading Coefficient: 1

Graph C:

$LaTeX: f\left(x\right)=5x^2-x$

Degree: 2

Leading Coefficient: 5

Screenshot 2024-11-15 at 12.24.03 PM.png

Screenshot 2024-11-15 at 12.23.35 PM.png

Screenshot 2024-11-15 at 12.23.03 PM.png

We can see that if the degree of the polynomial is even and the leading coefficient is negative, then the graph of the function extends down on both ends as in graphs D, E and F.

Graph D:

$LaTeX: f\left(x\right)=-3x^4+x$

Degree: 4

Leading Coefficient: -3

Graph E:

$LaTeX: f\left(x\right)=-x^4+3x^2-2$

Degree: 4

Leading Coefficient: -1

Graph F:

$LaTeX: f\left(x\right)=-x^2$

Degree: 2

Leading Coefficient: -1

Screenshot 2024-11-15 at 12.22.33 PM.png

Screenshot 2024-11-15 at 12.22.08 PM.png

Screenshot 2024-11-15 at 12.21.43 PM.png

Now consider the graphs below. Each of the polynomials show in the graphs have an odd degree.

We can see that if the degree of the polynomial is odd and the leading coefficient is positive, then the graph of the function extends up on the right and down on the left as in graphs A, B and C.

Graph A:

$LaTeX: f\left(x\right)=x^3-3x+2$

Degree: 3

Leading Coefficient: 1

Graph B:

$LaTeX: f\left(x\right)=2x^5-4x^3$

Degree: 5

Leading Coefficient: 2

Graph C:

$LaTeX: f\left(x\right)=0.5x^7$

Degree: 7

Leading Coefficient: 0.5

Screenshot 2024-11-15 at 12.21.14 PM.png

Screenshot 2024-11-15 at 12.20.44 PM.png

Screenshot 2024-11-15 at 12.20.16 PM.png

We can see that if the degree of the polynomial is odd and the leading coefficient is negative, then the graph of the function extends up on the left and down on the right as in graphs D, E and F.

Graph D:

$LaTeX: f\left(x\right)=-2x^3+x$

Degree: 3

Leading Coefficient: -2

Graph E:

$LaTeX: f\left(x\right)=-x^5+x^2$

Degree: 5

Leading Coefficient: -1

Graph F:

$LaTeX: f\left(x\right)=-5x^7$

Degree: 7

Leading Coefficient: -5

Screenshot 2024-11-15 at 12.19.39 PM.png

Screenshot 2024-11-15 at 12.19.15 PM.png

Screenshot 2024-11-15 at 12.18.47 PM.png

INSTRUCTIONS: Identify which of the following equations could be the function for the graph given. Consider the end behavior of the graph. Explain your answer.

Function A: $LaTeX: f\left(x\right)=2x^5-x$

Function B: $LaTeX: f\left(x\right)=-x^4+3x$

Function C: $LaTeX: f\left(x\right)=-4x^5+x$

Screenshot 2024-11-15 at 11.50.43 AM.png

Solution

Function C because the leading coefficient is negative and the degree is odd. This is consistent with the graph because the graph extends up to the left and down to the right.

INSTRUCTIONS: Consider the end behavior of each graph and match the graph to the possible function equation:

Function A: $LaTeX: f\left(x\right)=2x^4-x$

Function B: $LaTeX: f\left(x\right)=-x^3+3x$

Function C: $LaTeX: f\left(x\right)=-x^4+5x^2+3$

Graph 1:

Screenshot 2024-11-15 at 11.43.32 AM.png

Graph 2:

Screenshot 2024-11-15 at 11.44.14 AM.png

Graph 3:

Screenshot 2024-11-15 at 11.44.53 AM.png

Solution

Function A is Graph 3

Function B is Graph 1

Function C is Graph 2

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