6.4 Multiplying/Dividing Polynomials

1/10/2014

How do you multiply 1256 by 13?

Example 1 Multiply Polynomials Vertically

Find the product .( )x 2 4x 7–+ ( )2x –

SOLUTION

Line up like terms vertically. Then multiply as shown below.

x 2 4x 7–+

2x –×

2x 2 8x + 14–– Multiply by 2.x 2 4x 7–+ –

x 3 7x+ –4x 2 Multiply by x.x 2 4x 7–+

x 3 15x+ –2x 2 + 14 Combine like terms.

Example 2 Multiply Polynomials Horizontally

Find the product.

( )4+3x ( )5x 2 x 6–+a.

4+3x( )5x 2 x 6–+ ( )5x 2 x 6–+= Use distributive property.

SOLUTION

( )4+3x ( )5x 2 x 6–+a.

+15x 3 18x–+ 20x 2 4x 24–+= 3x 2 Use distributive property.

15x 3 + 24–= 20x 23x 2 + 18x 4x– + Group like terms.

15x 3 14x–+= 23x 2 24– Combine like terms.

Example 2 Multiply Polynomials Horizontally

( )2x – ( )x 2 2x 3–+=

b. ( )2x –( )1x – ( )3x +

+x 3 3x–+ 2x 2 4x 6–= 2x 2 – Use distributive property.

x 3 + 6= 2x 22x 2 3x 4x–– – + Group like terms.

+x 3 7x– 6= Combine like terms.

2x –( )x 2 2x 3–+= ( )x 2 2x 3–+

𝑥2+2𝑥− 3 Multiply any 2 binomials

Multiply by the 3rd binomial

Checkpoint Multiply Polynomials

Find the product. Use either a horizontal or vertical format.

1. ( )1+x( )x 2 x 2+ +

2. ( )2x 2 x 4– + ( )3x –

ANSWER 2x 2 3x 2x 3 + + +

ANSWER 7x 2 7x 122x 3 +– –

Checkpoint Multiply Polynomials

Find the product. Use either a horizontal or vertical format.

3. ( )1+2x( )3x 2 x 1–+

ANSWER 5x 2 x 16x 3 + ––

ANSWER x 2 10x 8x 3 + – +

( )1x –4. ( )4x + ( )2x –

Checkpoint Use Special Product Patterns

Find the product.

( )7z –( )7z +5.

ANSWER z 2 49–

6. ( )223y +

12y9y 2 4+ +ANSWER

64x 3 +– 12x 48x 2 – 1ANSWER

7. –( )314x

Homework:

Worksheet 6.4

6.4 Multiplying/Dividing Polynomials….cont’d.

1/29/2014

Example 4 Use Long Division

Find the quotient 985 23.÷

Divide 98 by 23.98523-92 Subtract the product .4( )23 = 92

65 Bring down 5. Divide 65 by 23.

19 Remainder

ANSWER The result is written as .2319

42

-46 Subtract the product .2( )23 = 46

4 2

Example 5 Use Polynomial Long Division

Find the quotient .( )4x +x 3 + –6x 3x 2 – 4 ( ) ÷

Write division in the same format you use to divide whole numbers.

x 3 + 4x 2 Subtract the product .( )4x +x 2 = x 3 4x 2+

– 6x x 2– Bring down - 6x. Divide –x2 by x

– 4x x 2– Subtract the product . ( )4x +x = x 2 4x – – –

– 2x – 4 Bring down - 4. Divide -2x by x

4 Remainder

x 3 + – 6x 3x 2 – 4 x + 4 x 3 ÷ x = x 2

ANSWER

The result is written as .x 2 – –x 2x + 4

4+

– 2x – 8 Subtract the product ( )4x +2 = 2x 8– – – .

x2 -x -2

- -

+ +

+ +

Example 5 Use Polynomial Long Division

CHECK You can check the result of a division problem by multiplying the divisor by the quotient and adding the remainder. The result should be the dividend (what’s inside ).

+ 4x 2 – –x 2( ) x + 4( ) = x + 4( )x 2 – x x + 4( ) – 2 x + 4( ) + 4

= x 3 – – 4x + 44x 2 x 2+ – 2x – 8

= x 3 – 6x3x 2+ – 4

Synthetic division:

Is a method of dividing polynomials by an expression of the form x - k

Example 1 Using Synthetic division

Divide: x – (-4) in x – k form

-4 Coefficients of powers of x1 3 -6 -4

k

1

-4

-1

4multiply

-2

8

4

add

coefficients of the power of x in descending order, starting with the power that is one less than that of the dividend.

ANSWER x 2 – –x 2x + 4

4+

remainder

Example 2 Using Synthetic division

Find the quotient:

3 Coefficients of powers of x2 -7 0 6 -14

k

2

6

-1

-3multiply

-3

-9

-23

add

remainder-3

-9

2 𝑥4− 7 𝑥3+0 𝑥2+6 𝑥− 14

ANSWER 𝟐 𝒙𝟑−𝒙𝟐−𝟑 𝒙−𝟑−𝟐𝟑𝒙−𝟑

Checkpoint

Find the quotient.

5x4x 3 +( )+ 1 x( + 1)÷

ANSWER

4x 4x2 +– 9 +x + 1

8–

Example 3Use Polynomial Long Division

Divide: Can’t use synthetic division because it isn’t being divided by x-k

2 𝑥2+7 𝑥− 392 𝑥−7 =

2 𝑥2 −7 𝑥- +

1 4 𝑥−39 =

𝑥

14 𝑥− 49- +

+7

1 0 remainder

𝐴𝑛𝑠𝑤𝑒𝑟 :𝑥+7+10

2 𝑥−7

Example 3 Use Polynomial Long Division

Divide:

Can’t use synthetic division because it isn’t being divided by x-k

2 𝑥4+3 𝑥3+0 𝑥2+5 𝑥− 1𝑥2−2 𝑥+2 =

2 𝑥2

2 𝑥4− 4 𝑥3+4 𝑥2- + -

+ =

+7 𝑥

+ - + -

1 0 𝑥2 −9 𝑥− 1 =

+10

1 0 𝑥2 −20 𝑥+20- + -

1 1𝑥− 21remainder

+

Homework:

WS: Dividing Polynomials