Section 4.2 Algebra of Polynomials

In the previous section, we defined functions and practiced using the compact function notation. In this section, we will explore a particularly useful kind of function.

In this section, you will....

define polynomials and explore their characteristics
evaluate polynomials
perform the four basic operations on polynomials

Supplemental Videos.

The main topics of this section are also presented in the following videos:

What is a Polynomial, How to Evaluate Polynomials, and Operations on Polynomials

A polynomial is an expression made up of a single term or sum of terms with only one variable in which each exponent is a positive whole number. For example,

\begin{equation*} 2x^3, \hspace{.5cm} 8x^4-x^2+9, \hspace{.5cm} x^9, \hspace{.5cm} x+4 \end{equation*}

are all polynomials, while

\begin{equation*} x^{-3}, \hspace{.5cm} 4x^{0.5}+2, \hspace{.5cm} \dfrac{1}{x^4} \end{equation*}

are not polynomials. Polynomials define functions of the form

\begin{gather*} f(x)=a_n x^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0. \end{gather*}

Here \(a_n\) represents any real number and \(n\) represents any whole number. The degree of a polynomial with one variable is the largest exponent of all the terms. Typically, we arrange terms of polynomials in descending order based on their degree and classify them as follows:

\begin{align*} \begin{aligned} f(x)\amp =2 \ \alert{\text{ Constant function (degree }0\text{)}}\\ g(x)\amp = 3x+2 \ \alert{\text{ Linear function (degree }1\text{)}}\\ h(x)\amp = 4x^2+3x+2 \ \alert{\text{ Quadratic function (degree }2\text{)}}\\ p(x)\amp = 5x^3+4x^2+3x+2 \ \alert{\text{ Cubic function (degree }3\text{)}} \end{aligned} \end{align*}

In this course, we call any polynomial with degree higher than 3 an \(n\)th-degree polynomial. For example, if the degree is 4, we call it a fourth-degree polynomial; if the degree is 5, we call it a fifth-degree polynomial, and so on.

When a polynomial is arranged in descending order based on their degree, we call the first term of the sum the leading term, and the coefficient part of this term is called the leading coefficient.

Example 4.14.

Given \(f(x)=3x^2+5x+17\text{,}\) \(g(x)=15x^{24}+36x^{22}+17x^4+31\text{,}\) \(h(x)=11x^2+17x^3+121x\text{,}\) \(j(x)=5x^3+2x+4x^3+3x+5\text{,}\) and \(k(x)=25\text{,}\) state the degree, leading term, and leading coefficient of each polynomial.

Solution.

Looking at \(f(x)\text{,}\) we see that the highest power of \(x\) is \(2\text{,}\) so the degree is \(2\text{.}\) The term with the highest degree is \(3x^2\text{,}\) meaning this is our leading term, and the coefficient of this term is \(3\text{,}\) which means this is our leading coefficient.

Looking at \(g(x)\text{,}\) we see that the highest power of \(x\) is \(24\text{,}\) so the degree is \(24\text{.}\) The term with the highest degree is \(15x^{24}\text{,}\) meaning this is our leading term, and the coefficient of this term is \(15\text{,}\) which means this is our leading coefficient.

Looking at \(h(x)\text{,}\) we see that the polynomial is not arranged in descending order of degree, so we will rearrange first.

\begin{equation*} h(x)=17x^3+11x^2+121x \end{equation*}

Now we see that the highest power of \(x\) is \(3\text{,}\) so the degree is \(3\text{.}\) The term with the highest degree is \(17x^{3}\text{,}\) meaning this is our leading term, and the coefficient of this term is \(17\text{,}\) which means this is our leading coefficient.

Looking at \(j(x)\text{,}\) we see that the polynomial is not arranged in descending order of degree, so we will rearrange first.

\begin{equation*} j(x)=5x^3+4x^3+2x+3x+5=9x^3+5x+5 \end{equation*}

Now we see that the highest power of \(x\) is \(3\text{,}\) so the degree is \(3\text{.}\) The term with the highest degree is \(9x^{3}\text{,}\) meaning this is our leading term, and the coefficient of this term is \(9\text{,}\) which means this is our leading coefficient.

Looking at \(k(x)\text{,}\) we have just a constant. This is the same as \(k(x)=25\cdot 1=25x^0\text{,}\) so we see that the highest power of \(x\) is \(0\text{,}\) so the degree is \(0\text{.}\) The term with the highest degree is \(25x^0=25\text{,}\) meaning this is our leading term, and the coefficient of this term is \(25\text{,}\) which means this is our leading coefficient.

Example 4.15.

Given \(f(x)=x^2-8x+17\text{,}\) find \(f(2)\) and \(f(4)\text{.}\)

Solution.

Replace each instance of \(x\) with the value given inside the parentheses.

\begin{equation*} \begin{aligned} f(\alert{2})\amp = (\alert{2})^2-8(\alert{2})+17\\ \amp =4-16+17\\ \amp =4+1\\ \amp =5 \end{aligned} \end{equation*}

\begin{equation*} \begin{aligned} f(\alert{4})\amp = (\alert{4})^2-8(\alert{4})+17\\ \amp =16-32+17\\ \amp =-16+17\\ \amp =1 \end{aligned} \end{equation*}

We can write \(f(2)=5\) and \(f(4)=1\text{.}\) Remember that \(f(x)=y\) and so we can interpret these results on the graph as follows:

Quadrants of the Plane

We find that \(f(2)=5\) and \(f(4)=1\text{.}\)

One useful application of polynomials appears in physics. The height of an object launched upward, ignoring the effects of air resistance, can be modeled with the following quadratic function:

\begin{gather*} h(t)=-\frac{1}{2}gt^2+v_0t+s_0 \end{gather*}

With this formula, the height \(h(t)\) can be calculated at any given time \(t\) after the object is launched. The letter \(g\) represents acceleration due to gravity on the surface of the Earth, which is 32 feet per second squared (or, using metric units, \(g = 9.8\) meters per second squared). The parameter \(v_0\) pronounced “v-naught,” or sometimes “v-zero,” represents the initial velocity of the object. The parameter \(s_0\) represents the initial height from which the object was launched.

Example 4.16.

An object is launched from the ground at a speed of \(64\) feet per second. Write a function that models the height of the object and use it to calculate the object’s height at \(1\) second and at \(3.5\) seconds.

Solution.

We know that the acceleration due to gravity is \(g=32\) feet per second squared and we are given the initial velocity \(v_0=64\) feet per second. Since the object is launched from the ground, the initial height is \(s_0=0\) feet. Create the mathematical model by substituting these coefficients into the following formula:

\begin{equation*} \begin{aligned} h(t)\amp =-\frac{1}{2}gt^2+v_ot+s_0\\ h(t)\amp = -\frac{1}{2}(\alert{32})t^2+(\alert{64})t+\alert{0}\\ h(t)\amp = -16t^2+64t \end{aligned} \end{equation*}

Use this model to calculate the height of the object at \(1\) second and \(3.5\) seconds.

\begin{equation*} \begin{aligned} h(\alert{1})\amp = -16(\alert{1})^2+64(\alert{1})=-16+64=48\\ h(\alert{3.5})\amp = -16(\alert{3.5})^2+64(\alert{3.5})=-196+224=28 \end{aligned} \end{equation*}

Our function is \(h(t)=-16t^2+64t\text{.}\) This tells us that at \(1\) second the object is at a height of \(48\) feet, and at \(3.5\) seconds it is at a height of \(28\) feet.

Example 4.17.

An object is dropped from a height of \(6\) meters. Write a function that models the height of the object and use it to calculate the object’s height \(1\) second after it is dropped.

Solution.

Our function is \(h(t)=-4.9t^2+6\text{.}\) Since

\begin{equation*} \begin{aligned} h(1) \amp= -4.9(1)^2+6 =1.1, \end{aligned} \end{equation*}

at \(1\) second the object is at a height of \(1.1\) meters.

Subsection 4.2.1 Adding and Subtracting Polynomials

On occasion a special notation is used to indicate addition and subtraction of functions. This mentioned here in the book so you have some exposure to the notation. The notation is as follows:

\begin{gather*} \text{Addition of functions: }(f+g)(x)=f(x)+g(x)\\ \text{Subtraction of functions: }(f-g)(x)=f(x)-g(x). \end{gather*}

When using function notation, be careful to group the entire function and add or subtract accordingly.

Example 4.18.

Given \(f(x)=x^3-5x-7\) and \(g(x)=3x^2+7x-2\text{,}\) find \(f(x)+g(x)\) and \(f(x)-g(x)\text{.}\)

Solution.

The notation \(f(x)+g(x)\) indicates that we should add the given expressions.

\begin{equation*} \begin{aligned} f(x)+g(x)\amp = (x^3-5x-7)+(3x^2+7x-2)\\ \amp = x^3-5x-7+3x^2+7x-2\\ \amp = x^3+3x^2+2x-9 \end{aligned} \end{equation*}

The notation \(f(x)-g(x)\) indicates that we should subtract the given expressions. When subtracting, the parentheses become very important. We can eliminate them after applying the distributive property.

\begin{equation*} \begin{aligned} f(x)-g(x) \amp = (x^3-5x-7)-(3x^2+7x-2)\\ \amp = x^3-5x-7-3x^2-7x+2\\ \amp = x^3-3x^2-12x-5 \end{aligned} \end{equation*}

Our solutions are \(f(x)+g(x)=x^3+3x^2+2x-9\) and \(f(x)-g(x)= x^3-3x^2-12x-5\text{.}\)

We may be asked to evaluate the sum or difference of two functions. We have the option to first find the sum or difference in general and then use the resulting function to evaluate for the given variable, or evaluate each first and then find the sum or difference.

Example 4.19.

Evaluate \(f(3)-g(3)\) given \(f(x)=5x^2-x+4\) and \(g(x)=x^2+2x-3\text{.}\)

Solution.

First, find \((f-g)(x)\text{.}\)

\begin{equation*} \begin{aligned} f(x)-g(x)\amp = (5x^2-x+4)-(x^2+2x-3)\\ \amp = 5x^2-x+4-x^2-2x+3\\ \amp = 4x^2-3x+7 \end{aligned} \end{equation*}

Now we substitute \(3\) in for the variable \(x\text{.}\)

\begin{equation*} \begin{aligned} f(\alert{3})-g(\alert{3}) \amp = 4(\alert{3})^2-3(\alert{3})+7\\ \amp = 36-9+7\\ \amp = 34 \end{aligned} \end{equation*}

Our solution is \(f(3)-g(3)=34\text{.}\)

Alternatively, we could have found \(f(3)\) and \(g(3)\) and then subtracted the results to obtain our answer.

Note: If multiple values are to be evaluated, it is best to find the sum or difference in general first and then use it to evaluate.

Example 4.20.

Given \(f(x)=x^3+x-8\) and \(g(x)=2x^2-x+9\text{,}\) evaluate \(f(2)+g(2)\) and \(f(-2)+g(-2)\text{.}\)

Solution.

We will compute \(f(x)+g(x)\) on its own first.

\begin{equation*} \begin{aligned} f(x)+g(x) \amp= (x^3+x-8)+(2x^2-x+9)=x^3+2x^2+1 \end{aligned} \end{equation*}

Using this, we see that

\begin{equation*} \begin{aligned} f(\alert{2})+g(\alert{2}) \amp= (\alert{2})^3+2(\alert{2})^2+1 \\ \amp= 8+2(4)+1 \\ \amp= 17 \end{aligned} \end{equation*}

and

\begin{equation*} \begin{aligned} f(\alert{-2})+g(\alert{-2}) \amp= (\alert{-2})^3+2(\alert{-2})^2+1 \\ \amp= -8+2(4)+1 \\ \amp= 1 \end{aligned} \end{equation*}

Subsection 4.2.2 Multiplying and Dividing Polynomials

The notation that is sometimes used to indicate multiplication and division of functions follows:

\begin{gather*} \text{Multiplication of functions: }(f\cdot g)(x)=f(x)\cdot g(x)\\ \text{Division of functions: }\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}, \text{ where }g(x)\neq 0. \end{gather*}

In this book we will simply consider multiplication notated by \(f(x)\cdot g(x)\) and division notated by \(\frac{f(x)}{g(x)}, \text{ where }g(x)\neq 0.\)

Example 4.21.

Given \(f(x)=15x^4-9x^3+6x^2\) and \(g(x)=3x^2\text{,}\) find \(f(x)\cdot g(x)\) and \(\frac{f(x)}{g(x)}\text{.}\)

Solution.

Apply the distributive property and simplify.

\begin{equation*} \begin{aligned} f(x)\cdot g(x)\amp = (15x^4-9x^3+6x^2)(3x^2)\\ \amp = 15x^4\cdot\alert{3x^2}-9x^3\cdot \alert{3x^2}+6x^2\cdot\alert{3x^2}\\ \amp = 45x^6-27x^5+18x^4 \end{aligned} \end{equation*}

For this quotient, assume \(x\neq 0\text{.}\)

\begin{equation*} \begin{aligned} \frac{f(x)}{g(x)}\amp = \frac{15x^4-9x^3+6x^2}{3x^2}\\ \amp = \frac{15x^4}{3x^2}-\frac{9x^3}{3x^2}+\frac{6x^2}{3x^2}\\ \amp = 5x^2-3x+2 \end{aligned} \end{equation*}

Our solutions are \(f(x)\cdot g(x)=45x^6-27x^5+18x^4\) and \(\frac{f(x)}{g(x)}= 5x^2-3x+2\text{.}\)

Example 4.22.

Given \(f(x)=6x-5\) and \(g(x)=3x^2-2x-1\text{,}\) find \(f(0)\cdot g(0)\) and \(f(-1)\cdot g(-1)\text{.}\)

Solution.

To find \(f(x)\cdot g(x)\text{,}\) we need to apply the distributive property multiple times. First, we distribute \(6x-5\) to each of the terms in \(g(x)\) and then distribute the terms to each of the \(6x-5\text{.}\)

\begin{equation*} \begin{aligned} f(x)\cdot g(x)\amp = (6x-5)(3x^2-2x-1) \\ \amp= (6x-5)(3x^2)-(6x-5)(2x)-(6x-5)(1) \\ \amp= (6x)(3x^2)-(5)(3x^2)-(6x)(2x)+(5)(2x)-(6x)(1)+(5)(1) \\ \amp= 18x^3-15x^2-12x^2+10x-6x+5 \\ \amp= 18x^3-27x^3+4x+5 \end{aligned} \end{equation*}

Now we substitute \(0\) and \(-1\) in for the variable \(x\text{.}\)

\begin{equation*} \begin{aligned} f(0)\cdot g(0)\amp = 18(0)^3-27(0)^2+4(0)+5\\ \amp =5 \end{aligned} \end{equation*}

\begin{equation*} \begin{aligned} f(-1)\cdot g(-1)\amp = 18(-1)^3-27(-1)^2+4(-1)+5\\ \amp = -18-27-4+5\\ \amp=-44 \end{aligned} \end{equation*}

Our solutions are \(f(0)\cdot g(0)=5\) and \(f(-1)\cdot g(-1)=-44\text{.}\)

Example 4.23.

Given \(f(x)=x^3+x-8\) and \(g(x)=2x^2-x+9\text{,}\) find \(f(x)\cdot g(x)\text{.}\)

Solution.

\begin{equation*} \begin{aligned} f(x)\cdot g(x) \amp= (x^3+x-8)(2x^2-x+9) \\ \amp= (x^3+x-8)(2x^2)-(x^3+x-8)(x)+(x^3+x-8)(9) \\ \amp= (x^3)(2x^2)+(x)(2x^2)-(8)(2x^2)-(x^3)(x)-(x)(x)\\ \amp \;\;\;\;\;\;\; +(8)(x)+(x^3)(9)+(x)(9)-(8)(9) \\ \amp= 2x^5+2x^3-16x^2-x^4-x^2+8x+9x^3+9x-72 \\ \amp= 2x^5-x^4+11x^3-17x^2+17x-72 \end{aligned} \end{equation*}

Exercises 4.2.3 Exercises

1. Degree and Coefficient of Monomials.

This is similar to Problem 1 from your Section 4.2 worksheet.

Give the coefficient and the degree of each term.

(a) \(-39 x^2\)

Coefficient:

Degree:

(b) \(26 x\)

Coefficient:

Degree:

Coefficient:

Degree:

(d) \(\frac{x}{3}\)

Coefficient:

Degree:

(e) \(27\)

Coefficient:

Degree:

Hint.

The coefficient is the number that the variable part is being multiplied. The degree of a term is the exponent on the term.

If there isn’t a coefficient, what number could you put in the coefficient’s place that would not change the expression? If there isn’t an exponent, what number could you put as an exponent that would change the expression?

If there is no variable part, how can you write the expression so that it has a variable part without changing the expressions itself? What exponent makes the variable part "go away"?

2. Degree and Coefficient of Polynomials.

This is similar to Problem 2 from your Section 4.2 worksheet.

Give the leading term, leading coefficient, and degree of each of the following polynomials.

(a) \(-17 x^7+18 x^2-3-x^5\)

Leading term: (make sure to use a lowercase "x")

Leading coefficient:

Degree:

(b) \(-4 y^3-3 y + 4 y^5+11\)

Leading term: (make sure to use a lowercase "y")

Leading coefficient:

Degree:

Leading term: (make sure to use a lowercase "x")

Leading coefficient:

Degree:

(d) \(11 y^6+15 y^7-y^5-15 y^4\)

Leading term: (make sure to use a lowercase "y")

Leading coefficient:

Degree:

Hint.

The leading term of a simplified polynomial is the term containing largest exponent. The leading coefficient of a polynomial is the coefficient of the leading term, and the degree of a polynomial is the degree of the leading term.

3. Adding and Subtracting Polynomials.

This is similar to Problem 3 from your Section 4.2 worksheet.

Let \begin{aligned} f(x) &= 8 x+6 x^3 - 6 \\ g(x) &= -2 x^4 + 3 x^3 - 7 \\ h(x) &= 4 x^2 + 9 x^4 - 6 \\ k(x) &= 9 x^3 - 3 x^4 - 4 x^2 \end{aligned} Add or subtract the polynomials as indicated. Simplify your answers completely, and be sure to use a lowercase "x".

(a) \(f(x)+g(x)=\)

(b) \(h(x)-k(x)=\)

(d) \(k(x)-f(x)=\)

Hint.

Be sure to put parentheses around the polynomials when you set up the addition and subtraction.

4. Evaluating Polynomials.

This is similar to Problem 4 from your Section 4.2 worksheet.

Evaluate the polynomial \(f(x)=9 x^3 - 3 x + 1\) for the values below.

\(f(-1)=\)

\(f(1)=\)

\(f(0)=\)

5. Evaluating Polynomials.

This is similar to Problem 4 from your Section 4.2 worksheet.

Evaluate the polynomial \(f(x)=-x^4+2 x^2-x+8\) for the values below.

\(f(-1)=\)

\(f(1)=\)

\(f(0)=\)

6. Adding, Subtracting, and Evaluating Polynomials.

This is similar to Problem 5 from your Section 4.2 worksheet.

Let \(f(x)=x^2-12\text{,}\) \(g(x)=-4 x\text{,}\) and \(h(x)=5 x-3\text{.}\) Find each of the following. Be sure to use a lowercase "x" when applicable.

\(f(x)-g(x)=\)

\(f(1)-g(1)=\)

\(h(x)-f(x)=\)

\(h(-2)-f(-2)=\)

\(g(x)+f(x)=\)

\(g(3)+f(3)=\)

\(h(x)+f(x)-g(x)=\)

\(h(-4)+f(-4)-g(-4)=\)

7. Selling T-Shirts.

This is similar to Problem 6 from your Section 4.2 worksheet.

The cost in dollars to produce \(s\) t-shirts is \(C(s)=5 s + 12\text{.}\) The revenue in dollars from sales of \(s\) t-shirts is \(R(s)=19 s\text{.}\) Write and simplify a function \(P\) that gives the profit for selling \(s\) t-shirts. Find the profit if \(49\) t-shirts are produced and sold.

The profit for selling \(s\) t-shirts is \(P(s)=\). (Be sure to simplify your answer and to use a lowercase "p")

The profit for selling \(49\) t-shirts is dollars.

Hint.

Profit is the amount of money made minus the amount of money spent in production costs.

\(P(s)=R(s)-C(s)\)