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# 9.1: Addition and Subtraction of Polynomials

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Write a polynomial expression in standard form.
• Classify polynomial expression by degree
• Problem solving using addition and subtraction of polynomials

## Introduction

So far we have seen functions described by straight lines (linear functions) and functions where the variable appeared in the exponent (exponential functions). In this section we will introduce polynomial functions. A polynomial is made up of different terms that contain positive integer powers of the variables. Here is an example of a polynomial.

\begin{align*}4x^3 + 2x^2 - 3x + 1\end{align*}

Each part of the polynomial that is added or subtracted is called a term of the polynomial. The example above is a polynomial with four terms.

The numbers appearing in each term in front of the variable are called the coefficients. The number appearing all by itself without a variable is called a constant.

In this case, the coefficient of \begin{align*}x^3\end{align*} is 4, the coefficient of \begin{align*}x^2\end{align*} is 2, the coefficient of \begin{align*}x\end{align*} is -3 and the constant is 1.

## Degrees of Polynomials and Standard Form

Each term in the polynomial has a degree. This is the power of the variable in that term.

\begin{align*}4x^3\end{align*} Has a degree of 3 and is called a cubic term or \begin{align*}3^{rd}\end{align*} order term.

\begin{align*}2x^2\end{align*} Has a degree of 2 and is called the quadratic term or \begin{align*}2^{nd}\end{align*} order term.

\begin{align*}-3x\end{align*} Has a degree of 1 and is called the linear term or \begin{align*}1^{st}\end{align*} order term.

1 Has a degree of 0 and is called the constant.

By definition, the degree of the polynomial is the same as the degree of the term with the highest degree. This example is a polynomial of degree 3, which is also called a “cubic” polynomial. (Why do you think it is called a cubic?).

Polynomials can have more than one variable. Here is another example of a polynomial.

\begin{align*}t^{4}-6s^{3}t^{2}-12st+4s^{4}-5\end{align*}

This is a polynomial because all exponents on the variables are positive integers. This polynomial has five terms. Let’s look at each term more closely. Note: The degree of a term is the sum of the powers on each variable in the term.

\begin{align*}t^{4}\end{align*} Has a degree of 4, so it’s a \begin{align*}4^{th}\end{align*} order term

\begin{align*}-6s^{3}t^{2}\end{align*} Has a degree of 5, so it’s a \begin{align*}5^{th}\end{align*} order term.

\begin{align*}-12^{st}\end{align*} Has a degree of 2, so it’s a \begin{align*}2^{nd}\end{align*} order term

\begin{align*}4s^{4}\end{align*} Has a degree of 4, so it’s a \begin{align*}4^{th}\end{align*} order term

-5 Is a constant, so its degree is 0.

Since the highest degree of a term in this polynomial is 5, then this is polynomial of degree 5 or a \begin{align*}5^{th}\end{align*} order polynomial.

A polynomial that has only one term has a special name. It is called a monomial (mono means one). A monomial can be a constant, a variable, or a product of a constant and one or more variables. You can see that each term in a polynomial is a monomial. A polynomial is the sum of monomials. Here are some examples of monomials.

\begin{align*}\underbrace{b^2}_{{\color{blue}\ \text{This is a monomial}}} && \underbrace{8}_{{\color{blue} \ \text{So is this}}} && \underbrace{-2ab^2}_{{\color{blue} \ \text{and this}}} && \underbrace{\frac{1} {4}x^4}_{{\color{blue} \ \text{and this}}} && \underbrace{-29xy}_{{\color{blue} \ \text{and this}}}\end{align*}

Example 1

For the following polynomials, identify the coefficient on each term, the degree of each term and the degree of the polynomial.

a) \begin{align*}x^5 - 3x^3 + 4x^2 - 5x + 7 \end{align*}

b) \begin{align*}x^4 - 3x^3 y^2 + 8x - 12\end{align*}

Solution

a) \begin{align*}x^5 - 3x^3 + 4x^2 - 5x + 7 \end{align*}

The coefficients of each term in order are 1, -3, 4, -5 and the constant is 7.

The degrees of each term are 5, 3, 2, 1, and 0. Therefore, the degree of the polynomial is 5.

b) \begin{align*}x^4 - 3x^3 y^2 + 8x - 12\end{align*}

The coefficients of each term in order are 1, -3, 8 and the constant is -12.

The degrees of each term are 4, 5, 1, and 0. Therefore, the degree of the polynomial is 5.

Example 2

Identify the following expressions as polynomials or non-polynomials.

a) \begin{align*} 5x^2 - 2x\end{align*}

b) \begin{align*} 3x^2 - 2x^{-2}\end{align*}

c) \begin{align*} x \sqrt{x} - 1\end{align*}

d) \begin{align*} \frac{5} {x^3 + 1}\end{align*}

e) \begin{align*} 4x^{\frac{1}{3}}\end{align*}

f) \begin{align*} 4xy^2 - 2x^2y - 3 + y^3 - 3x^3\end{align*}

Solution

(a) \begin{align*} 5x^2 - 2x\end{align*} This is a polynomial.

(b) \begin{align*} 3x^2 - 2x^{-2}\end{align*} This is not a polynomial because it has a negative exponent.

(c) \begin{align*} x \sqrt{x} - 1\end{align*} This is not a polynomial because is has a square root.

(d) \begin{align*} \frac{5} {x^3 + 1}\end{align*} This is not a polynomial because the power of \begin{align*}x\end{align*} appears in the denominator.

(e) \begin{align*} 4x^{\frac{1}{3}}\end{align*} This is not a polynomial because it has a fractional exponent.

(f) \begin{align*} 4xy^2 - 2x^y - 3 + y^3 - 3x^3\end{align*} This is a polynomial.

You saw that each term in a polynomial has a degree. The degree of the highest term is also the degree of the polynomial. Often, we arrange the terms in a polynomial so that the term with the highest degree is first and it is followed by the other terms in order of decreasing power. This is called standard form.

The following polynomials are in standard form.

\begin{align*}4x^4 - 3x^3 + 2x^2 - x + 1 \end{align*}

\begin{align*}a^4 b^3 - 2a^3 b^3 + 3a^4 b - 5ab^2 + 2\end{align*}

The first term of a polynomial in standard form is called the leading term and the coefficient of the leading term is called the leading coefficient.

The first polynomial above has a leading term of \begin{align*}4x^4\end{align*} and a leading coefficient of 4.

The second polynomial above has a leading term of \begin{align*}a^4 b^3\end{align*} and a leading coefficient of 1.

Example 3

Rearrange the terms in the following polynomials so that they are in standard form. Indicate the leading term and leading coefficient of each polynomial.

(a) \begin{align*}7 - 3x^3 + 4x\end{align*}

(b) \begin{align*}ab - a^3 + 2b\end{align*}

(c) \begin{align*}-4b + 4 + b^2\end{align*}

Solution

(a) \begin{align*}7 - 3x^3 + 4x\end{align*} is rearranged as \begin{align*}-3x^3 + 4x + 7\end{align*}. The leading term is \begin{align*}-3x^3\end{align*} and the leading coefficient is -3.

(b) \begin{align*}ab - a^3 + 2b\end{align*} is rearranged as \begin{align*}-a^3 + ab + 2b\end{align*}. The leading term is \begin{align*}-a^3\end{align*} and the leading coefficient is -1.

(c) \begin{align*}-4b + 4 +b^2\end{align*} is rearranged as \begin{align*}b^2 -4b + 4\end{align*}. The leading term is \begin{align*}b^2\end{align*} and the leading coefficient is 1.

## Simplifying Polynomials

A polynomial is simplified if it has no terms that are alike. Like terms are terms in the polynomial that have the same variable(s) with the same exponents, but they can have different coefficients.

\begin{align*}2x^2y\end{align*} and \begin{align*}5x^2y\end{align*} are like terms.

\begin{align*}6x^2y\end{align*} and \begin{align*}6xy^2\end{align*} are not like terms.

If we have a polynomial that has like terms, we simplify by combining them.

\begin{align*}& x^2 + \underline{6xy} - \underline{4xy} + y^2\\ & \qquad \nearrow \qquad \nwarrow\\ & \qquad \text{Like terms}\end{align*}

This sample polynomial simplified by combining the like terms \begin{align*}6xy - 4xy = 2xy\end{align*}. We write the simplified polynomial as

\begin{align*}x^2 + 2xy + y^2\end{align*}

Example 4

Simplify the following polynomials by collecting like terms and combining them.

(a) \begin{align*}2x - 4x^2 + 6 + x^2 -4 + 4x\end{align*}

(b) \begin{align*}a^3b^3 - 5ab^4 + 2a^3b - a^3b^3 + 3ab^4 - a^2b\end{align*}

Solution

(a) \begin{align*}2x - 4x^2 + 6 + x^2 -4 + 4x\end{align*}

Rearrange the terms so that like terms are grouped together

\begin{align*}=(-4x^2 + x^2) + (2x + 4x) + (6 -4)\end{align*}

Combine each set of like terms by adding or subtracting the coefficients

\begin{align*}= -3x^2 + 6x + 2\end{align*}

(b) \begin{align*}a^3b^3 - 5ab^4 + 2a^3b - a^3b^3 + 3ab^4 - a^2b\end{align*}

Rearrange the terms so that like terms are grouped together:

\begin{align*}= (a^3b^3 - a^3b^3) + (-5ab^4 + 3ab^4) + 2a^3b - a^2b\end{align*}

Combine each set of like terms:

\begin{align*}= 0 - 2ab^4 + 2a^3b - a^2b \!\\ = - 2ab^4 + 2a^3b - a^2b\end{align*}

To add two or more polynomials, write their sum and then simplify by combining like terms.

Example 5

Add and simplify the resulting polynomials.

(a) Add \begin{align*}3x^2 - 4x + 7\end{align*} and \begin{align*}2x^3 - 4x^2 - 6x + 5.\end{align*}

(b) Add \begin{align*}x^2 - 2xy + y^2\end{align*} and \begin{align*}2y^2 - 4x^2\end{align*} and \begin{align*}10xy + y^3.\end{align*}

Solution:

(a) Add \begin{align*}3x^2 - 4x + 7\end{align*} and \begin{align*}2x^3 - 4x^2 - 6x + 5\end{align*}

\begin{align*} &=(3x^2 - 4x + 7) + (2x^3 - 4x^2 - 6x + 5)\\ \text{Group like terms} &= 2x^3 + (3x^2 - 4x^2) + (-4x - 6x) + (7 + 5)\\ \text{Simplify} &= 2x^3 - x^2 - 10x +12\end{align*}

(b) Add \begin{align*}x^2 - 2xy + y^2\end{align*} and \begin{align*}2y^2 - 3x^2\end{align*} and \begin{align*}10xy + y^3\end{align*}

\begin{align*}& =(x^2 - 2xy + y^2) + (2y^2 - 3x^2) + (10xy + y^3)\\ \text{Group like terms} & = (x^2 - 3x^2) + (y^2 + 2y^2) + (-2xy + 10xy) + y^3\\ \text{Simplify} & = 2x^2 +3y^2 + 8xy + y^3\end{align*}

Polynomial subtraction

To subtract one polynomial from another, add the opposite of each term of the polynomial you are subtracting.

Example 6

Subtract and simplify the resulting polynomials.

a) Subtract \begin{align*}x^3 - 3x^2 + 8x + 12\end{align*} from \begin{align*}4x^2 + 5x- 9.\end{align*}

b) Subtract \begin{align*}5b^2 - 2a^2\end{align*} from \begin{align*}4a^2 - 8ab - 9b^2.\end{align*}

Solution

a) \begin{align*}(4x^2 + 5x - 9) - (x^3 - 3x^2 + 8x + 12) & = (4x^2 + 5x - 9) + (-x^3 + 3x^2 - 8x -12)\\ \text{Group like terms} & = - x^3 - (4x^2 + 3x^2) + (5x - 8x) + (-9 - 12)\\ \text{Simplify} & = - x^3 + 7x^2 - 3x -21\end{align*}

b) \begin{align*}(4a^2 - 8ab - 9b^2) - (5b^2 - 2a^2) & = (4a^2 - 8ab - 9b^2) + (-5b^2 + 2a^2)\\ \text{Group like terms} & = (4a^2 + 2a^2) + (-9b^2 - 5b^2)- 8ab\\ \text{Simplify} & = 6a^2 - 14b^2 - 8ab\end{align*}

Note: An easy way to check your work after adding or subtracting polynomials is to substitute a convenient value in for the variable, and check that your answer and the problem both give the same value. For example, in part (b) of Example 6, if we let \begin{align*}a = 2\end{align*} and \begin{align*}b = 3\end{align*}, then we can check as follows.

\begin{align*}& \text{Given} && \text{Solution}\\ & (4a^2 - 8ab - 9b^2) - (5b^2 - 2a^2) && 6a^2 - 14b^2 - 8ab \\ & (4(2)^2 - 8(2)(3) - 9(3)^2) - (5(3)^2 - 2(2)^2) && 6(2)^2 - 14(3)^2 - 8(2)(3) \\ & (4(4) - 8(2)(3) - 9(9)) - (5(9) - 2(4)) && 6(4) - 14(9) - 8(2)(3) \\ & (-113) - 37 && 24 - 126 - 48 \\ & -150 && -150\end{align*}

Since both expressions evaluate to the same number when we substitute in arbitrary values for the variables, we can be reasonably sure that our answer is correct. Note, when you use this method, do not choose 0 or 1 for checking since these can lead to common problems.

## Problem Solving Using Addition or Subtraction of Polynomials

An application of polynomials is their use in finding areas of a geometric object. In the following examples, we will see how the addition or subtraction of polynomials might be useful in representing different areas.

Example 7

Write a polynomial that represents the area of each figure shown.

a)

b)

c)

d)

Solutions

a) This shape is formed by two squares and two rectangles.

The blue square has area: \begin{align*}y \cdot y = y^2 \end{align*}

The yellow square has area: \begin{align*}x \cdot x = x^2 \end{align*}

The pink rectangles each have area: \begin{align*}x \cdot y = xy\end{align*}

To find the total area of the figure we add all the separate areas.

\begin{align*}\text{Total area} & = y^2 + x^2 + xy + xy\\ & = y^2 + x^2 + 2xy\end{align*}

b) This shape is formed by two squares and one rectangle.

The yellow squares each have an area: \begin{align*}a \cdot a = a^2.\end{align*}

The orange rectangle has area: \begin{align*}2a \cdot b = 2ab.\end{align*}

To find the total area of the figure we add all the separate areas.

\begin{align*}\text{Total area} & = a^2 + a^2 + 2ab\\ & = 2a^2 + 2ab\end{align*}

c) To find the area of the green region we find the area of the big square and subtract the area of the little square.

The big square has area \begin{align*}y \cdot y = y^2. \end{align*}

The little square has area \begin{align*}x \cdot x = x^2.\end{align*}

Area of the green region \begin{align*}= y^2 - x^2\end{align*}

d) To find the area of the figure we can find the area of the big rectangle and add the areas of the pink squares.

The pink squares each have area: \begin{align*}a \cdot a = a^2.\end{align*}

The blue rectangle has area: \begin{align*}3a \cdot a = 3a^2.\end{align*}

To find the total area of the figure we add all the separate areas.

\begin{align*}\text{Total area} = a^2 + a^2 + a^2 + 3a^2 = 6a^2\end{align*}

Another way to find this area is to find the area of the big square and subtract the areas of the three yellow squares.

The big square has area: \begin{align*}3a \cdot 3a = 9a^2.\end{align*}

The yellow squares each have areas: \begin{align*}a \cdot a = a^2.\end{align*}

To find the total area of the figure we subtract:

\begin{align*}\text{Area} & = 9a^2 - (a^2 + a^2 + a^2)\\ & = 9a^2 - 3a^2\\ & = 6a^2\end{align*}

## Review Questions

Indicate which expressions are polynomials.

1. \begin{align*} x^2 + 3x^{\frac{1}{2}}\end{align*}
2. \begin{align*} \frac{1} {3}x^2y - 9y^2\end{align*}
3. \begin{align*} 3x^{-3}\end{align*}
4. \begin{align*} \frac{2} {3}t^2 - \frac{1} {t^2}\end{align*}

Express each polynomial in standard form. Give the degree of each polynomial.

1. \begin{align*}3 - 2x\end{align*}
2. \begin{align*}8 - 4x + 3x^3\end{align*}
3. \begin{align*}-5 + 2x - 5x^2 + 8x^3\end{align*}
4. \begin{align*}x^2 - 9x^4 + 12\end{align*}

1. \begin{align*}(x + 8) + (-3x - 5)\end{align*}
2. \begin{align*}(-2x^2 + 4x -12) + (7x + x^2)\end{align*}
3. \begin{align*}(2a^2b - 2a + 9) + (5a^2b - 4b + 5)\end{align*}
4. \begin{align*}(6.9a^2 - 2.3b^2 + 2ab) + (3.1a - 2.5b^2 + b)\end{align*}

Subtract and simplify.

1. \begin{align*}(-t + 15t^2) - (5t^2 + 2t - 9)\end{align*}
2. \begin{align*}(-y^2 + 4y - 5) - (5y^2 + 2y + 7)\end{align*}
3. \begin{align*}(-5m^2 - m) - (3m^2 + 4m - 5)\end{align*}
4. \begin{align*}(2a^2b - 3ab^2 + 5a^2b^2) - (2a^2b^2 +4a^2b - 5b^2)\end{align*}

Find the area of the following figures.

1. No
2. yes
3. no
4. no
5. \begin{align*}- 2x + 3; \ \text{Degree} = 1\end{align*}
6. \begin{align*}3x^3 - 4x + 8; \ \text{Degree} = 3\end{align*}
7. \begin{align*}8x^3 - 5x^2 + 2x - 5; \ \text{Degree} = 3\end{align*}
8. \begin{align*}-9x^4 + x^2 + 12; \ \text{Degree} = 4\end{align*}
9. \begin{align*}-2x + 3\end{align*}
10. \begin{align*}-x^2 + 11x - 12\end{align*}
11. \begin{align*}7a^2b - 2a - 4b + 14\end{align*}
12. \begin{align*}6.9a^2 - 4.8b^2 + 2ab + 3.1a + b\end{align*}
13. \begin{align*}-3t + 9\end{align*}
14. \begin{align*}-6y^2 + 2y - 12\end{align*}
15. \begin{align*}-8m^2 - 5m + 5\end{align*}
16. \begin{align*}-2a^2b - 3ab^2 + 3a^2b^2 + 5b^2\end{align*}
17. \begin{align*}\text{Area} = 2xz - xy\end{align*}
18. \begin{align*}\text{Area} = 4ab + ac\end{align*}
19. \begin{align*}2xy - 2x^2\end{align*}
20. \begin{align*}\text{Area} = 3ab\end{align*}

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