Adding and Subtracting Polynomials

 

Learning Objectives

·         Add polynomials.

·         Find the opposite of a polynomial.

·         Subtract polynomials.

 

Introduction

 

Adding and subtracting polynomials may sound complicated, but it’s really not much different from the addition and subtraction that you do every day. The main thing to remember is to look for and combine like terms.

 

Adding Polynomials

 

You can add two (or more) polynomials as you have added algebraic expressions. You can remove the parentheses and combine like terms.

 

 

Example

Problem

Add. (3b + 5) + (2b + 4)

 

(3b + 2b) + (5 + 4)

Regroup using the commutative property of addition and the associative property of addition.

 

5b + 9

Combine like terms.

Answer

(3b + 5) + (2b + 4) = 5b + 9

 

 

 

Example

Problem

A rectangular garden has one side with a length of x + 7 and another with a length 2x + 3. Find the perimeter of the garden.

 

 

(x + 7) + (2x + 3) + (x + 7) + (2x + 3)

The perimeter of a rectangle is the sum of its side lengths.

 

(x + 2x + x + 2x) + (7 + 3 + 7 + 3)

Regroup by like terms using commutative and associative properties.

 

 

6x + 20

 

Add like terms.

 

Answer

 

The perimeter is 6x + 20.

 

 

 

The procedure is the same when you add polynomials that contain negative coefficients or subtraction:

 

 

Example

Problem

Add. (-5x2 – 10x + 2) + (3x2 + 7x – 4)

 

-5x2 + (-10x) + 2 + 3x2 + 7x + (-4)

Rewrite subtraction as addition of the opposite.

 

 

(-5x2 + 3x2) + (-10x + 7x) + (2 – 4)

Regroup using commutative and associative properties.

 

 

-2x2 + (-3x) + (-2)

 

Combine like terms.

Answer

(−5x2 – 10x + 2) + (3x2 + 7x – 4) = −2x2 – 3x – 2

 

 

The above examples show addition of polynomials horizontally, by reading from left to right along the same line. Some people like to organize their work vertically instead, because they find it easier to be sure that they are combining like terms. The example below shows this “vertical” method of adding polynomials:

 

 

Example

Problem

Add. (3x2 + 2x – 7) + (7x2 – 4x + 8)

 

 

 

 

3x2

+

2x

–

7

 

+

7x2

–

4x

+

8

 

Write one polynomial below the other, making sure to line up like terms.

 

 

3x2

+

2x

–

7

 

+

7x2

–

4x

+

8

 

 

10x2

–

2x

+

1

 

Combine like terms, paying close attention to the signs.

Answer

(3x2 + 2x – 7) + (7x2 – 4x + 8) = 10x2 – 2x + 1

 

 

Sometimes in a vertical arrangement, you can line up every term beneath a like term, as in the example above. But sometimes it isn't so tidy. When there isn't a matching like term for every term, there will be empty places in the vertical arrangement.

 

 

Example

Problem

Add. (4x3 + 5x2 – 6x + 2) + (−4x2 + 10)

 

 

 

 

4x3

+

5x2

–

6x

+

2

+

 

 

−4x2

 

 

+

10

Write one polynomial below the other, lining up like terms vertically.

 

Leave a blank space above or below every term without a matching like term.

 

 

4x3

+

5x2

–

6x

+

2

+

 

 

-4x2

 

 

+

10

 

 

4x3

+

x2

–

6x

+

12

Combine like terms, paying close attention to the signs.

Answer

(4x3 + 5x2 – 6x + 2) + (−4x2 + 10) = 4x3 + x2 – 6x + 12

 

 

Find the sum.

(4a2 + 5a + 7) + (8a + 2)

 

A) 9a2 + 8a + 9

 

B) 16a2 + 10a

 

C) 12a2 + 5a + 9

 

D) 4a2 + 13a + 9

 

Show/Hide Answer

A) 9a2 + 8a + 9

Incorrect. 4a2 and 5a cannot be combined because they are not like terms. Regroup by like terms to get 4a2 + (5a + 8a) + (7 + 2). The correct answer is 4a2 + 13a + 9.

 

B) 16a2 + 10a

Incorrect. You must first regroup by like terms before you add coefficients. Regrouping gives 4a2 + (5a + 8a) + (7 + 2). The correct answer is 4a2 + 13a + 9.

 

C) 12a2 + 5a + 9

Incorrect. You can’t combine 4a2 and 8a because they aren’t like terms. Regroup by like terms to get 4a2 + (5a + 8a) + (7 + 2). The correct answer is 4a2 + 13a + 9.

 

D) 4a2 + 13a + 9

Correct. Regroup by like terms to get 4a2 + (5a + 8a) + (7 + 2), which is 4a2 + 13a + 9.

 

 

 

Finding the Opposite of a Polynomial

 

When you subtract polynomials, you will add the opposite, as you have done with real numbers. So how do you find the opposite of a polynomial? Recall that the opposite of 3 is −3, and the opposite of −3 is 3. Just as the opposite of a number is found by multiplying the number by −1, we can find the opposite of a polynomial by multiplying it by −1.

 

 

Example

Problem

Find the opposite of 9x2 + 10x + 5.

 

(−1)(9x2 + 10x + 5)

Find the opposite by multiplying by −1.

 

(−1)9x2 + (−1)10x + (−1)5

 

Distribute −1 to each term in the polynomial.

 

−9x2 + (-10x) + (−5)

 

Multiply each coefficient by −1.

Answer

The opposite of 9x2 + 10x + 5 is −9x2 – 10x – 5.

Rewrite addition of a negative term as subtraction.

 

 

Be careful when there are negative values or subtractions in the polynomial already.

 

 

Example

Problem

Find the opposite of 3p2 – 5p + 7.

 

(-1)(3p2 – 5p + 7)

Find the opposite by multiplying by -1.

 

(-1)[3p2 + (-5)p + 7]

Change the subtraction to adding the opposite.

 

(-1)3p2 + (-1)(-5)p + (-1)7

 

Distribute -1 to each term in the polynomial and multiply each coefficient by -1.

 

-3p2 + 5p + (-7)

Rewrite addition of a negative term as subtraction.

Answer

The opposite of 3p2 – 5p + 7 is −3p2 + 5p – 7.

 

 

Notice that in finding the opposite of a polynomial, you change the sign of each term in the polynomial.

 

 

Find the opposite of the polynomial.

8a3 – 3a – 2

 

A) -8a3 – 3a – 2

 

B) 8a3 + 3a – 2

 

C) 8a3 + 3a + 2

 

D) -8a3 + 3a + 2

 

Show/Hide Answer

A) -8a3 – 3a – 2

Incorrect. Each term must have the sign changed. The correct answer is -8a3 + 3a + 2.

 

B) 8a3 + 3a – 2

Incorrect. Each term must have the sign changed. The correct answer is -8a3 + 3a + 2.

 

C) 8a3 + 3a + 2

Incorrect. Each term must have the sign changed. The correct answer is -8a3 + 3a + 2.

 

D) -8a3 + 3a + 2

Correct. Each term must have the sign changed, so the answer is -8a3 + 3a + 2.

 

 

 

Subtracting Polynomials

 

Just as subtracting real numbers is the same as adding the opposite, you can subtract polynomials by adding the opposite of the second polynomial. Let's look at an example:

 

Example

Problem

Subtract. (15x2 + 12x + 20) – (9x2 + 10x + 5)

 

(15x2 + 12x + 20) + (-9x2 – 10x – 5)

Change the subtraction to adding the opposite. Be careful to change the sign of each term!

 

(15x2 + -9x2) + (12x – 10x) + (20 – 5)

Regroup to match like terms.

 

6x2  + 2x + 15

Combine like terms.

Answer

(15x2 + 12x + 20) – (9x2 + 10x + 5) = 6x2  + 2x + 15

 

 

When polynomials include a lot of terms, it can be easy to lose track of the signs. Be careful to transfer them correctly, especially when subtracting a negative term.

 

 

Example

Problem

Subtract. (14x3 + 3x2  – 5x + 14) – (7x3 + 5x2  – 8x + 10)

 

(14x3 + 3x2 – 5x + 14) + (-7x3 – 5x2 + 8x – 10)

Rewrite as adding the opposite.

 

14x3 + 3x2 + (-5)x + 14 + (-7)x3 + (-5)x2 + 8x +(-10)

You may want to rewrite all subtractions as adding the opposite.

 

14x3 + (-7)x3 + 3x2 + (-5)x2 + (-5)x + 8x + 14 + (-10)

Regroup to put like terms together.

 

7x3 + (-2)x2 + 3x + 4

Combine like terms.

Answer

(14x3 + 3x2  – 5x + 14) – (7x3 + 5x2  – 8x + 10) = 7x3 – 2x2 + 3x + 4

 

 

Complex problems, like the one above, may be more easily solved using the vertical approach (shown below). However you choose to combine polynomials is up to you—the key point is to identify like terms, and be able to organize them accurately.

 

 

Example

Problem

Subtract. (14x3  + 3x2  – 5x + 14) – (7x3 + 5x2  – 8x + 10)

 

 

   14x3

+

3x2

–

5x

+

  14

– (7x3

+

5x2

–

8x

+

10)

Reorganizing using the vertical approach.

 

 

 

14x3

+

3x2

–

5x

+

14

 

 

-7x3

–

5x2

+

8x

–

10

 

 

7x3

–

2x2

+

3x

+

4

Change the subtraction to adding the opposites, and combine like terms.

Answer

(14x3 + 3x2  – 5x + 14) – (7x3 + 5x2  – 8x + 10) = 7x3 – 2x2 + 3x + 4

 

 

As with integer operations, experience and practice makes it easier to add and subtract polynomials.

 

 

Subtract.

(4a3 – 5a + 7) – (8a3 – 3a – 2)

 

A) -4a3 – 8a + 5

 

B) --4a3 + 3a + 9

 

C) -4a3 – 2a + 9

 

D) -4a3 + 2a + 5

 

Show/Hide Answer

A) -4a3 – 8a + 5

Incorrect. Each term in the second polynomial must be subtracted from the first polynomial: 4a3 – 5a + 7 – 8a3 + 3a + 2. The correct answer is -4a3 – 2a + 9.

 

B) -4a3 + 3a + 9

Incorrect. The subtracted 5a in the first polynomial must still be subtracted: 4a3 – 5a + 7 – 8a3 + 3a + 2. The correct answer is -4a3 – 2a + 9.

 

C) -4a3 – 2a + 9

Correct. 4a3 – 5a + 7 – 8a3 + 3a + 2 = -4a3 – 2a + 9.

 

D) -4a3 + 2a + 5

Incorrect. Each term in the second polynomial must be subtracted from the first polynomial: 4a3 – 5a + 7 – 8a3 + 3a + 2. The correct answer is -4a3 – 2a + 9.

 

 

 

Summary

 

When adding or subtracting polynomials, use the commutative and associative properties to regroup the terms in a polynomial into groups of like terms. Change subtraction, including subtraction of the second polynomial, to addition of the opposite. When finding the opposite of a polynomial, be sure to change the sign of each term. Then you can combine the like terms.