<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

# Addition and Subtraction of Polynomials

## Combining like terms in polynomial expressions

Estimated14 minsto complete
%
Progress
Practice Addition and Subtraction of Polynomials
Progress
Estimated14 minsto complete
%

What if you had two polynomials like 4x25\begin{align*}4x^2 - 5\end{align*} and 13x+2\begin{align*}13x + 2\end{align*}? How could you add and subtract them? After completing this Concept, you'll be able to perform addition and subtraction on polynomials like these.

### Try This

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!)

### Watch This

CK-12 Foundation: 0902S Lesson Addition and Subtraction of Polynomials

### Guidance

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

#### Example A

Add and simplify the resulting polynomials.

a) Add 3x24x+7\begin{align*}3x^2-4x+7\end{align*} and 2x34x26x+5\begin{align*}2x^3-4x^2-6x+5\end{align*}

b) Add x22xy+y2\begin{align*}x^2-2xy+y^2\end{align*} and 2y23x2\begin{align*}2y^2-3x^2\end{align*} and 10xy+y3\begin{align*}10xy+y^3\end{align*}

Solution

a)

Group like terms:Simplify:(3x24x+7)+(2x34x26x+5)=2x3+(3x24x2)+(4x6x)+(7+5)=2x3x210x+12

b)

Group like terms:Simplify:(x22xy+y2)+(2y23x2)+(10xy+y3)=(x23x2)+(y2+2y2)+(2xy+10xy)+y3=2x2+3y2+8xy+y3

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

#### Example B

a) Subtract x33x2+8x+12\begin{align*}x^3-3x^2+8x+12\end{align*} from 4x2+5x9\begin{align*}4x^2+5x-9\end{align*}

b) Subtract 5b22a2\begin{align*}5b^2-2a^2\end{align*} from 4a28ab9b2\begin{align*}4a^2-8ab-9b^2\end{align*}

Solution

a)

(4x2+5x9)(x33x2+8x+12)Group like terms:Simplify:=(4x2+5x9)+(x3+3x28x12)=x3+(4x2+3x2)+(5x8x)+(912)=x3+7x23x21

b)

(4a28ab9b2)(5b22a2)Group like terms:Simplify:=(4a28ab9b2)+(5b2+2a2)=(4a2+2a2)+(9b25b2)8ab=6a214b28ab

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 a=2\begin{align*}a=2\end{align*} and b=3\begin{align*}b=3\end{align*}, then we can check as follows:

Given(4a28ab9b2)(5b22a2)(4(2)28(2)(3)9(3)2)(5(3)22(2)2)(4(4)8(2)(3)9(9))(5(9)2(4))(113)37150Solution6a214b28ab6(2)214(3)28(2)(3)6(4)14(9)8(2)(3)2412648150

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 C

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.

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

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

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

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.

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.

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

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

To find the total area of the figure we subtract:

Watch this video for help with the Examples above.

CK-12 Foundation: Addition and Subtraction of Polynomials

### Guided Practice

Subtract \begin{align*} 4t^2+7t^3-3t-5\end{align*} from \begin{align*}6t+3-5t^3+9t^2\end{align*}.

Solution:

When subtracting polynomials, we have to remember to subtract each term. If the term is already negative, subtracting a negative term is the same thing as adding:

The final answer is in standard form.

### Explore More

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*}
5. \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.

1. \begin{align*}(-t+5t^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*}
5. \begin{align*}(3.5x^2y-6xy+4x)-(1.2x^2y-xy+2y-3)\end{align*}

Find the area of the following figures.

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 9.2.

### Vocabulary Language: English

distributive property

distributive property

The distributive property states that the product of an expression and a sum is equal to the sum of the products of the expression and each term in the sum. For example, $a(b + c) = ab + ac$.
Polynomial

Polynomial

A polynomial is an expression with at least one algebraic term, but which does not indicate division by a variable or contain variables with fractional exponents.