# 9.1: Addition and Subtraction of Polynomials

**At Grade**Created by: CK-12

## Learning Objectives

- Write a polynomial expression in standard form.
- Classify polynomial expression by degree.
- Add and subtract polynomials.
- Solve problems using addition and subtraction of polynomials.

## Introduction

So far we’ve seen functions described by straight lines (linear functions) and functions where the variable appeared in the exponent (exponential functions). In this section we’ll 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 different **degree**. The degree of the term is the power of the variable in that term.

\begin{align*}& 4x^3 && \text{has degree} \ 3 \ \text{and is called a cubic term or} \ 3^{rd} \ \text{order term}.\\ & 2x^2 && \text{has degree} \ 2 \ \text{and is called a quadratic term or} \ 2^{nd} \ \text{order term}.\\ & -3x && \text{has degree} \ 1 \ \text{and is called a linear term or} \ 1^{st} \ \text{order term}.\\ & 1 && \text{has degree} \ 0 \ \text{and is called the constant}.\end{align*}

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

This is a polynomial because all the 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.* In other words, the degree of each term is the number of variables that are multiplied together in that term, whether those variables are the same or different.

\begin{align*}& t^4 && \text{has a degree of} \ 4, \ \text{so it’s a} \ 4^{th} \ \text{order term}\\ & -6s^3t^2 && \text{has a degree of} \ 5, \ \text{so it’s a} \ 5^{th} \ \text{order term}.\\ & -12st && \text{has a degree of} \ 2, \ \text{so it’s a} \ 2^{nd} \ \text{order term}.\\ & 4s^4 && \text{has a degree of} \ 4, \ \text{so it’s a} \ 4^{th} \ \text{order term}.\\ & -5 && \text{is a constant, so its degree is} \ 0.\end{align*}

Since the highest degree of a term in this polynomial is 5, then this is polynomial of degree \begin{align*}5^{th}\end{align*} 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, so a polynomial is just the sum of several monomials. Here are some examples of monomials:

\begin{align*}b^2 \qquad -2ab^2 \qquad 8 \qquad \frac{1}{4}x^4 \qquad -29xy\end{align*}

**Example 1**

*For the following polynomials, identify the coefficient of each term, the constant, 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^3y^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, and -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^3y^2+8x-12\end{align*}

The coefficients of each term in order are 1, -3, and 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^5-2x\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) This ** is** a polynomial.

b) This is ** not** a polynomial because it has a negative exponent.

c) This is ** not** a polynomial because it has a radical.

d) This is ** not** a polynomial because the power of \begin{align*}x\end{align*} appears in the denominator of a fraction (and there is no way to rewrite it so that it does not).

e) This is ** not** a polynomial because it has a fractional exponent.

f) This ** is** a polynomial.

Often, we arrange the terms in a polynomial 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^4b^3-2a^3b^3+3a^4b-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 the leading term \begin{align*}4x^4\end{align*}, and the leading coefficient is 4.

The second polynomial above has the leading term \begin{align*}a^4b^3\end{align*}, and the leading coefficient is 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*} becomes \begin{align*}-3x^3+4x+7\end{align*}. Leading term is \begin{align*}-3x^3\end{align*}; leading coefficient is -3.

b) \begin{align*}ab-a^3+2b\end{align*} becomes \begin{align*}-a^3+ab+2b\end{align*}. Leading term is \begin{align*}-a^3\end{align*}; leading coefficient is -1.

c) \begin{align*}-4b+4+b^2\end{align*} becomes \begin{align*}b^2-4b+4\end{align*}. Leading term is \begin{align*}b^2\end{align*}; 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, whether they have the same or different coefficients.

For example, \begin{align*}2x^2y\end{align*} and \begin{align*}5x^2y\end{align*} are like terms, but \begin{align*}6x^2y\end{align*} and \begin{align*}6xy^2\end{align*} are not like terms.

When a polynomial has like terms, we can simplify it by combining those terms.

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

We can simplify this polynomial by combining the like terms \begin{align*}6xy\end{align*} and \begin{align*}-4xy\end{align*} into \begin{align*}(6-4)xy\end{align*}, or \begin{align*}2xy\end{align*}. The new polynomial is \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) 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: \begin{align*}-3x^2+6x+2\end{align*}

b) 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*}

## Adding and Subtracting Polynomials

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

**Solution**

a) \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) \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*}

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

**Example 6**

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) above, 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

One way we can use polynomials is to find the area of a geometric figure.

**Example 7**

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

a)

b)

c)

d)

**Solution**

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

\begin{align*}\text{The blue square has area} \ y \times y & = y^2.\\ \text{The yellow square has area} \ x \times x & = x^2.\\ \text{The pink rectangles each have area} \ x \times y & = xy.\end{align*}

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

\begin{align*}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.

\begin{align*}\text{The yellow squares each have area} \ a \times a & = a^2.\\ \text{The orange rectangle has area} \ 2a \times b & = 2ab.\end{align*}

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

\begin{align*}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.

\begin{align*}\text{The big square has area}: y \times y & = y^2.\\ \text{The little square has area}: x \times x & = x^2.\\ Area \ of \ the \ green \ region & = 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.

\begin{align*}\text{The pink squares each have area} \ a \times a & = a^2.\\ \text{The blue rectangle has area} \ 3a \times a & = 3a^2.\end{align*}

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

\begin{align*}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:

\begin{align*}\text{The big square has area} \ 3a \times 3a & = 9a^2.\\ \text{The yellow squares each have area} \ a \times a & = a^2.\end{align*}

To find the total area of the figure we subtract:

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

## Further Practice

For more practice adding and subtracting polynomials, try playing the Battleship game at http://www.quia.com/ba/28820.html. (The problems get harder as you play; watch out for trick questions!)

## Review Questions

Indicate whether each expression is a polynomial.

- \begin{align*}x^2+3x^{\frac{1}{2}}\end{align*}
- \begin{align*}\frac{1}{3}x^2y-9y^2\end{align*}
- \begin{align*}3x^{-3}\end{align*}
- \begin{align*}\frac{2}{3}t^2-\frac{1}{t^2}\end{align*}
- \begin{align*}\sqrt{x}-2x\end{align*}
- \begin{align*}\left ( x^\frac{3}{2} \right )^2\end{align*}

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

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

Add and simplify.

- \begin{align*}(x+8)+(-3x-5)\end{align*}
- \begin{align*}(-2x^2+4x-12)+(7x+x^2)\end{align*}
- \begin{align*}(2a^2b-2a+9)+(5a^2b-4b+5)\end{align*}
- \begin{align*}(6.9a^2-2.3b^2+2ab)+(3.1a-2.5b^2+b)\end{align*}
- \begin{align*}\left ( \frac{3}{5}x^2-\frac{1}{4}x+4 \right )+ \left ( \frac{1}{10}x^2 + \frac{1}{2}x-2\frac{1}{5} \right )\end{align*}

Subtract and simplify.

- \begin{align*}(-t+5t^2)-(5t^2+2t-9)\end{align*}
- \begin{align*}(-y^2+4y-5)-(5y^2+2y+7)\end{align*}
- \begin{align*}(-5m^2-m)-(3m^2+4m-5)\end{align*}
- \begin{align*}(2a^2b-3ab^2+5a^2b^2)-(2a^2b^2+4a^2b-5b^2)\end{align*}
- \begin{align*}(3.5x^2y-6xy+4x)-(1.2x^2y-xy+2y-3)\end{align*}

Find the area of the following figures.

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