3.2 Dividing Polynomials11/28/2012

Review: Quotient of Powers

Ex.

In general:

xx

xxxxx

x

x

2

5

nmn

m

aa

a

253 xx

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 1 Use Polynomial Long Division

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

- -

+ +

+ +

Divide:

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

Evaluate when x = -4:

k

= -64 + 48 + 24 – 4 = -16 + 24 – 4 = 8 – 4 = 4

Isn’t this the remainder when we performed synthetic division?

Remainder Thm: If a polynomials f(x) is divided by x – k, then the remainder is r = f(k)

Example 2 Using Synthetic division and Remainder Theorem

Evaluate using synthetic division and Remainder Thm:

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

k

2

6

-1

-3multiply

-3

-9

-23

add

remainder-3

-9

P(3)= -23

𝑃 (𝑥 )=2𝑥4 −7 𝑥3+0 𝑥2+6 𝑥−14

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:

Worksheet 3.2 #1-5all, 11-19odd, 23-

25all