ALGEBRA 2

Section 6.5:The Remainder and Factor Theorems

Polynomial Long Division

Use the same steps with polynomial long division as

with integers

Example 1

Integer Division Polynomial Division

786923 7235 2 xxx

3

69

x3

xx 153 2

969249

42

463

23

3

x17 7

17

8517x78

5

78

x

5

78173

xx

Divide 7 by 2 Divide 3x2 by x

Subtract 69Subtract 3x2 +15x

(change signs)

Remainder of 3

Remainder of 78

Example 2

02026202 2342 xxxxxx

23x

234 606 xxx23 62 xx x2

x

xxx 202 23

xx 46 20

3

606 2 xx

64x

22

642x

x

1

3233

2

2

x

xxx

)22()226( 234 xxxx

Reduce the fraction!

Pull a negative out of the top

Recall: Long Division Example 1

7235 2 xxx 5

78173

xx

See what happens when

you substitute 5 into the dividend

723)( 2 xxxf

7)5(2)5(3)5( 2f

71075)5(f

78)5(fAlso, remember that synthetic substitution

Gave us another way to evaluate the function

7235

3

15

17

85

78

remainder

Value of f(5)

Remainder Theorem

As we saw in the last example:

If a polynomial f(x) is divided by x – k, then the

remainder is the same as f(k).

Synthetic Division

Is used to divide a polynomial by a binomial of the

form x – k. (degree must be 1)

Example 1:

Divide using synthetic division

36137 24 xxxx

0613073

Opposite sign!

3

70223476217 23

xxxx

7022286321

70223476217

Assignment

p.356

#20-38 evens

(10 problems)

Journal:

Logical Reasoning

p.357 #59

Get into a groups of three

1. One person needs to factor the polynomial listed below.

2. Another person need to divide the polynomial by x – 2.

3. The last person needs evaluate f(2).

Compare your results.

3( ) 2 9 18f x x x x

1. (x - 2)(x + 3)(x - 3)

2. x2 - 9

3. f(2) = 0

What do these results mean?

A polynomial f(x) has a factor of x - k iff f(k) = 0.

Example: Factor the polynomial given that f(k) = 0.

3 23 13 2 8; 4x x x k

13

3 21. ( ) 3 13 2 8; 4f x x x x k

4

12 4 8

3 1 2 0

3 13 2 8

2( 4)(3 2)x x x

( 4)(3 2)( 1)x x x

First use synthetic division to factor given that -4 is one zero(this also meansx+4 is a factor)

Rewrite the polynomial inFactored form

Factor the quadraticusing trial & error

14

Given one zero of the polynomial, find the other zeros.

3 22. ( ) 6 3 10; 5f x x x x

15

3 22. ( ) 6 3 10; 5f x x x x

5

5 5 10

1 1 2 0

1 6 3 10

2( 5)( 2) 0x x x

( 5)( 2)( 1) 0x x x

5 2 1x x x

First use synthetic division to factor given that -5 is one zero

Rewrite the polynomial inFactored form(and set equal to zero)

Factor the quadraticusing trial & error

Solve each factor, set each equal to zero

5 0, 2 0, 1 0x x x

16

Given one zero of the polynomial, find the other zeros.

3 23. ( ) 14 47 18; 9f x x x x

17

9

9 45 18

1 5 2 0

1 14 47 18

2( 9)( 5 2)x x x

3 23. ( ) 14 47 18; 9f x x x x

18

2( 9)( 5 2)x x x

5 179,

2x

2 4

2

b b acx

a

2( 5) ( 5) 4(1)(2)

2(1)x

5 25 8

2x

5 17

2

Use the quadratic formula!

p.357#40-58 evens(10 problems)

Journal:Fuel Consumptionp.357 #61