Operations on Polynomials

OPERATIONS ON

POLYNOMIALS

Like TermsLike Terms

Like Terms refers to monomials that have the same variable(s) but may have different coefficients. The variables in the terms must have the same powers.

Which terms are like? 3a2b, 4ab2, 3ab, -5ab2

4ab2 and -5ab2 are like.

Even though the others have the same variables, the exponents are not the same.

3a2b = 3aab, which is different from 4ab2 = 4abb.

Constants are like terms.

Which terms are like? 2x, -3, 5b, 0

-3 and 0 are like.

Which terms are like? 3x, 2x2, 4, x

3x and x are like.

Which terms are like? 2wx, w, 3x, 4xw

2wx and 4xw are like.

Add: (x2 + 3x + 1) + (4x2 +5)

Step 1: Underline like terms:

Step 2: Add the coefficients of like terms, do not change the powers of the variables:

Adding PolynomialsAdding Polynomials

(x2 + 3x + 1) + (4x2 +5)

Notice: ‘3x’ doesn’t have a like term.

(x2 + 4x2) + 3x + (1 + 5)

5x2 + 3x + 6

Some people prefer to add polynomials by stacking them. If you choose to do this, be sure to line up the like terms!

(x2 + 3x + 1) + (4x2 +5)

5x2 + 3x + 6

(x2 + 3x + 1) + (4x2 +5)

Stack and add these polynomials: (2a2+3ab+4b2) + (7a2+ab+-2b2)

(2a2+3ab+4b2) + (7a2+ab+-2b2)

(2a2 + 3ab + 4b2)

+ (7a2 + ab + -2b2)

9a2 + 4ab + 2b2

1) 3x3 7x 3x3 4x 6x3 3x

2) 2w2 w 5 4w2 7w 1 6w2 8w 4

3) 2a3 3a2 5a a3 4a 3 3a3 3a2 9a 3

• Add the following polynomials; you may stack them if you prefer:

Subtract: (3x2 + 2x + 7) - (x2 + x + 4)

Subtracting PolynomialsSubtracting Polynomials

Step 1: Change subtraction to addition (Keep-Change-Change.).

Step 2: Underline OR line up the like terms and add.

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

(3x2 + 2x + 7)

+ (- x2 + - x + - 4)

2x2 + x + 3

1) x2 x 4 3x 2 4x 1 2x2 3x 5

2) 9y2 3y 1 2y2 y 9 7y2 4y 10

3) 2g2 g 9 g3 3g2 3 g3 g2 g 12

• Subtract the following polynomials by changing to addition (Keep-Change-Change.), then add:

Multiplication

of Polynomials

Warm UpEvaluate.

1. 32

3. 102

Simplify.

4. 23 24

6. (53)2

9 16

100

27

2. 24

5. y5 y4

56 7. (x2)4

8. –4(x – 7) –4x + 28

y9

x8

To multiply monomials and polynomials, you will use some of the properties of exponents that you learned earlier in this chapter.

Multiply.

Example 1: Multiplying Monomials

A. (6y3)(3y5)

(6y3)(3y5)

18y8

Group factors with like bases together.

B. (3mn2) (9m2n)

(3mn2)(9m2n)

27m3n3

Multiply.


Multiply.

(6 3)(y3 y5)

(3 9)(m m2)(n2 n)

Multiply.

Example 1C: Multiplying Monomials

2 2 2112

4s t st st

4 53s t


Multiply.

( )( )æçè

- 22 21

124

ts tt s sö÷ø

( )( )æ-

öçè

2 21

124

t s ts ts÷ø

2

When multiplying powers with the same base, keep the base and add the exponents.

x2 x3 = x2+3 = x5

Remember!

Check It Out! Example 1

Multiply.

a. (3x3)(6x2)

(3x3)(6x2)

(3 6)(x3 x2)18x5


Multiply.


Multiply.

b. (2r2t)(5t3)

(2r2t)(5t3)

(2 5)(r2)(t3 t)

10r2t4


Multiply.


Multiply.

c.

( )( )æçè

4 52 21

123

x zy zx yö÷ø

3

( )( )æçè

21

123

x y x z y z2 4 53ö÷ø

( )( )( )g gg gæçè

3 22 4 51

12 z3

zx x y yö÷ø

7554 x y z

To multiply a polynomial by a monomial, use the Distributive Property.

Multiply.

Example 2A: Multiplying a Polynomial by a Monomial

4(3x2 + 4x – 8)

4(3x2 + 4x – 8)

(4)3x2 +(4)4x – (4)8

12x2 + 16x – 32

Distribute 4.

Multiply.

6pq(2p – q)

(6pq)(2p – q)

Multiply.

Example 2B: Multiplying a Polynomial by a Monomial

(6pq)2p + (6pq)(–q)

(6 2)(p p)(q) + (–1)(6)(p)(q q)

12p2q – 6pq2

Distribute 6pq.

Group like bases together.

Multiply.

Multiply.

Example 2C: Multiplying a Polynomial by a Monomial


Multiply.

( )+21

62

x y xy x y8 2 2

x y( )+ 22 61

2xyyx 8 2

x y x y( ) ( )æçè

+2 21

62

8xy x y 22ö÷ø

1

2

æçè

ö÷ø

x2 • x( )( ) ( )( )æ+ç

è

1• 6

2y • y x2 • x2 y • y2• 8

ö÷ø

æçè

ö÷ø

1

2

3x3y2 + 4x4y3

Distribute .21x y

2


Multiply.

a. 2(4x2 + x + 3)

2(4x2 + x + 3)

2(4x2) + 2(x) + 2(3)

8x2 + 2x + 6

Distribute 2.

Multiply.


Multiply.

b. 3ab(5a2 + b)

3ab(5a2 + b)

(3ab)(5a2) + (3ab)(b)

(3 5)(a a2)(b) + (3)(a)(b b)

15a3b + 3ab2

Distribute 3ab.


Multiply.


Multiply.

c. 5r2s2(r – 3s)

5r2s2(r – 3s)

(5r2s2)(r) – (5r2s2)(3s)

(5)(r2 r)(s2) – (5 3)(r2)(s2 s)

5r3s2 – 15r2s3

Distribute 5r2s2.


Multiply.

To multiply a binomial by a binomial, you can apply the Distributive Property more than once:

(x + 3)(x + 2) = x(x + 2) + 3(x + 2) Distribute x and 3.

Distribute x and 3 again.

Multiply.

Combine like terms.

= x(x + 2) + 3(x + 2)

= x(x) + x(2) + 3(x) + 3(2)

= x2 + 2x + 3x + 6

= x2 + 5x + 6

Another method for multiplying binomials is called the FOIL method.

4. Multiply the Last terms. (x + 3)(x + 2) 3 2 = 6

3. Multiply the Inner terms. (x + 3)(x + 2) 3 x = 3x

2. Multiply the Outer terms. (x + 3)(x + 2) x 2 = 2x

F

O

I

L

(x + 3)(x + 2) = x2 + 2x + 3x + 6 = x2 + 5x + 6

F O I L

1. Multiply the First terms. (x + 3)(x + 2) x x = x2

Multiply.

Example 3A: Multiplying Binomials

(s + 4)(s – 2)

(s + 4)(s – 2)

s(s – 2) + 4(s – 2)

s(s) + s(–2) + 4(s) + 4(–2)

s2 – 2s + 4s – 8

s2 + 2s – 8

Distribute s and 4.

Distribute s and 4 again.

Multiply.

Combine like terms.

Multiply.

Example 3B: Multiplying Binomials

(x – 4)2

(x – 4)(x – 4)

(x x) + (x (–4)) + (–4 x) + (–4 (–4))

x2 – 4x – 4x + 8

x2 – 8x + 8

Write as a product of two binomials.

Use the FOIL method.

Multiply.

Combine like terms.

Example 3C: Multiplying Binomials

Multiply.

(8m2 – n)(m2 – 3n)

8m2(m2) + 8m2(–3n) – n(m2) – n(–3n)

8m4 – 24m2n – m2n + 3n2

8m4 – 25m2n + 3n2


Multiply.

Combine like terms.

In the expression (x + 5)2, the base is (x + 5). (x + 5)2 = (x + 5)(x + 5)

Helpful Hint

Check It Out! Example 3a

Multiply.

(a + 3)(a – 4)

(a + 3)(a – 4)

a(a – 4)+3(a – 4)

a(a) + a(–4) + 3(a) + 3(–4)

a2 – a – 12

a2 – 4a + 3a – 12

Distribute a and 3.

Distribute a and 3 again.

Multiply.

Combine like terms.

Check It Out! Example 3b

Multiply.

(x – 3)2

(x – 3)(x – 3)

(x x) + (x(–3)) + (–3 x)+ (–3)(–3) ●

x2 – 3x – 3x + 9

x2 – 6x + 9

Write as a product of two binomials.


Multiply.

Combine like terms.

Check It Out! Example 3c

Multiply.

(2a – b2)(a + 4b2)

(2a – b2)(a + 4b2)

2a(a) + 2a(4b2) – b2(a) + (–b2)(4b2)

2a2 + 8ab2 – ab2 – 4b4

2a2 + 7ab2 – 4b4


Multiply.

Combine like terms.

To multiply polynomials with more than two terms, you can use the Distributive Property several times. Multiply (5x + 3) by (2x2 + 10x – 6):

(5x + 3)(2x2 + 10x – 6) = 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)

= 5x(2x2 + 10x – 6) + 3(2x2 + 10x – 6)

= 5x(2x2) + 5x(10x) + 5x(–6) + 3(2x2) + 3(10x) + 3(–6)

= 10x3 + 50x2 – 30x + 6x2 + 30x – 18

= 10x3 + 56x2 – 18

You can also use a rectangle model to multiply polynomials with more than two terms. This is similar to finding the area of a rectangle with length (2x2 + 10x – 6) and width (5x + 3):

2x2 +10x –6

10x3 50x2 –30x

30x6x2 –18

5x

+3

Write the product of the monomials in each row and column:

To find the product, add all of the terms inside the rectangle by combining like terms and simplifying if necessary.

10x3 + 6x2 + 50x2 + 30x – 30x – 18

10x3 + 56x2 – 18

Another method that can be used to multiply polynomials with more than two terms is the vertical method. This is similar to methods used to multiply whole numbers.

2x2 + 10x – 6

5x + 3

6x2 + 30x – 18+ 10x3 + 50x2 – 30x

10x3 + 56x2 + 0x – 18

10x3 + 56x2 – 18

Multiply each term in the top polynomial by 3.

Multiply each term in the top polynomial by 5x, and align like terms.

Combine like terms by adding vertically.

Simplify.

Multiply.

Example 4A: Multiplying Polynomials

(x – 5)(x2 + 4x – 6)

(x – 5 )(x2 + 4x – 6)

x(x2 + 4x – 6) – 5(x2 + 4x – 6)

x(x2) + x(4x) + x(–6) – 5(x2) – 5(4x) – 5(–6)

x3 + 4x2 – 5x2 – 6x – 20x + 30

x3 – x2 – 26x + 30

Distribute x and –5.

Distribute x and −5 again.

Simplify.

Combine like terms.

Multiply.

Example 4B: Multiplying Polynomials

(2x – 5)(–4x2 – 10x + 3)

(2x – 5)(–4x2 – 10x + 3)

–4x2 – 10x + 32x – 5x

20x2 + 50x – 15

+ –8x3 – 20x2 + 6x

–8x3 + 56x – 15

Multiply each term in the top polynomial by –5.


Combine like terms by adding vertically.

Multiply.

Example 4C: Multiplying Polynomials

(x + 3)3

[(x + 3)(x + 3)](x + 3)

[x(x+3) + 3(x+3)](x + 3)

(x2 + 3x + 3x + 9)(x + 3)

(x2 + 6x + 9)(x + 3)

Write as the product of three binomials.

Use the FOIL method on the first two factors.

Multiply.

Combine like terms.

Example 4C: Multiplying Polynomials

Multiply.

(x + 3)3

x3 + 6x2 + 9x + 3x2 + 18x + 27

x3 + 9x2 + 27x + 27

x(x2) + x(6x) + x(9) + 3(x2) + 3(6x) + 3(9)

x(x2 + 6x + 9) + 3(x2 + 6x + 9)

Use the Commutative Property of Multiplication.

Distribute the x and 3.

Distribute the x and 3 again.

(x + 3)(x2 + 6x + 9)

Combine like terms.

Example 4D: Multiplying Polynomials

Multiply.

(3x + 1)(x3 – 4x2 – 7)

x3 4x2 –7

3x4 12x3 –21x

4x2x3 –7

3x

+1

3x4 + 12x3 + x3 + 4x2 – 21x – 7

Write the product of the monomials in each row and column.

Add all terms inside the rectangle.

3x4 + 13x3 + 4x2 – 21x – 7 Combine like terms.

A polynomial with m terms multiplied by a polynomial with n terms has a product that, before simplifying has mn terms. In Example 4A, there are 2 3, or 6 terms before simplifying.

Helpful Hint

Check It Out! Example 4a

Multiply.

(x + 3)(x2 – 4x + 6)

(x + 3 )(x2 – 4x + 6)

x(x2 – 4x + 6) + 3(x2 – 4x + 6)

Distribute x and 3.

Distribute x and 3 again.

x(x2) + x(–4x) + x(6) +3(x2) +3(–4x) +3(6)

x3 – 4x2 + 3x2 +6x – 12x + 18

x3 – x2 – 6x + 18

Simplify.

Combine like terms.

Check It Out! Example 4b

Multiply.

(3x + 2)(x2 – 2x + 5)

(3x + 2)(x2 – 2x + 5)

x2 – 2x + 5 3x + 2

Multiply each term in the top polynomial by 2.


2x2 – 4x + 10

+ 3x3 – 6x2 + 15x

3x3 – 4x2 + 11x + 10Combine like terms by adding

vertically.

Example 5: ApplicationThe width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height.

a. Write a polynomial that represents the area of the base of the prism.

Write the formula for the area of a rectangle.

Substitute h – 3 for w and h + 4 for l.

A = l w

A = l w

A = (h + 4)(h – 3)

Multiply.A = h2 + 4h – 3h – 12

Combine like terms.A = h2 + h – 12

The area is represented by h2 + h – 12.

Example 5: ApplicationThe width of a rectangular prism is 3 feet less than the height, and the length of the prism is 4 feet more than the height.

b. Find the area of the base when the height is 5 ft.

A = h2 + h – 12

A = h2 + h – 12

A = 52 + 5 – 12

A = 25 + 5 – 12

A = 18

Write the formula for the area the base of the prism.

Substitute 5 for h.

Simplify.

Combine terms.

The area is 18 square feet.


The length of a rectangle is 4 meters shorter than its width.

a. Write a polynomial that represents the area of the rectangle.

Write the formula for the area of a rectangle.

Substitute x – 4 for l and x for w.

A = l w

A = l w

A = x(x – 4)

Multiply.A = x2 – 4x

The area is represented by x2 – 4x.


The length of a rectangle is 4 meters shorter than its width.

b. Find the area of a rectangle when the width is 6 meters.

A = x2 – 4x

A = x2 – 4x

A = 36 – 24

A = 12

Write the formula for the area of a rectangle whose length is 4 meters shorter than width .

Substitute 6 for x.

Simplify.

Combine terms.

The area is 12 square meters.

A = 62 – 4 6

Lesson Quiz: Part I

Multiply.

1. (6s2t2)(3st)

2. 4xy2(x + y)

3. (x + 2)(x – 8)

4. (2x – 7)(x2 + 3x – 4)

5. 6mn(m2 + 10mn – 2)

6. (2x – 5y)(3x + y)

4x2y2 + 4xy3

18s3t3

x2 – 6x – 16

2x3 – x2 – 29x + 28

6m3n + 60m2n2 – 12mn

6x2 – 13xy – 5y2

Lesson Quiz: Part II

7. A triangle has a base that is 4cm longer than its height.

a. Write a polynomial that represents the area of the triangle.

b. Find the area when the height is 8 cm.

48 cm2

1

2h2 + 2h

Education

Operations on Polynomials