6.5 6.5 The Remainder and Factor TheoremsThe Remainder and Factor Theorems

Do Now:Do Now: Use long division to divide. Round to the nearest Use long division to divide. Round to the nearest hundredth.hundredth.

1) 4759 divided by 361) 4759 divided by 36 2) 12 divided by 52) 12 divided by 5

OBJ:OBJ: use long division to divide polynomialsuse long division to divide polynomials

Polynomial Long DivisionPolynomial Long DivisionIf a term is missing in standard form, fill it in with a 0 If a term is missing in standard form, fill it in with a 0

coefficient.coefficient.

Ex 1:Ex 1:

2x2x44 + 3x + 3x33 + 5x – 1 + 5x – 1 xx22 – 2x + 2 – 2x + 2

1503222 2342 xxxxxx

2x4 = 2x2

x2

2x2

1503222 2342 xxxxxx

2x2

+4x2-4x32x4-( )

- 4x27x3 +5x

7x3 = 7x x2

+7x

7x3 - 14x2 +14x-( )

10x2 - 9x -1

+10

10x2 - 20x +20-( )

11x - 21

remainder

The answer is written:The answer is written:

Quotient + Remainder over Quotient + Remainder over DivisorDivisor

2x2x22 + 7x + 10 + + 7x + 10 + 11x – 2111x – 21 x x22 – 2x + 2 – 2x + 2

yy44 + 2y + 2y22 – y + 5 – y + 5 yy22 – y + 1 – y + 1

yy22 + y + 2 + + y + 2 + 3 3 y y22 – y + – y +

11

Ex 2Ex 2::

Remainder TheoremRemainder Theorem::

If a polynomial is divisible by (x – If a polynomial is divisible by (x – kk), then ), then the the remainder remainder is f(is f(kk))

HW:

ONLY when you divide by x-k,

check if the remainder is f(k) by directly substituting k back into the eqn!!!

On your homework, this is possible on problems #6 and # 7

Synthetic DivisionSynthetic Division (like synthetic substitution)(like synthetic substitution)

Ex 3:Ex 3:

Divide Divide f(x)= 3xf(x)= 3x33 – 2x – 2x22 + 2x – 5 by x - 2 + 2x – 5 by x - 2

continued…

Do Now: Do Now: Use long division to divide Use long division to divide f(x)= 3xf(x)= 3x33 – 2x – 2x22 + 2x – 5 + 2x – 5 by by x x - 2- 2

3x3x2 2 + 4x + 10 + + 4x + 10 + 15 15 x – x –

22

f(x)= f(x)= 33xx33 – 2– 2xx22 + + 22x x – 5– 5 Divide by Divide by x - 2x - 2

33 -2-2 22 -5-522

3

6

4

8

10

20

R: 15

33xx22 + + 44xx + + 1010 + + 1515 x-2x-2

Divide xDivide x33 + 2x + 2x22 – 6x -9 by – 6x -9 by x-2x-2

11 22 -6-6 -9-9 22

11

2

44

8

22

4

R:-5

xx22 + 4x + 2 + 4x + 2 + + -5 -5 x-2 x-2

Ex 4:Ex 4:

Divide xDivide x33 + 2x + 2x22 – 6x -9 by – 6x -9 by x+3x+3

11 22 -6-6 -9-9

-3-3

1

-3

-1

3

-3

9

R: 0

xx22 – x - 3 – x - 3

Ex 5:Ex 5:

Factor Theorem:Factor Theorem:A polynomial has the A polynomial has the factor factor x-k if x-k if f(k)=0f(k)=0

** if the remainder is zero, x-k is a factor ** if the remainder is zero, x-k is a factor since it divides evenlysince it divides evenly

** it also means that k is a “zero” (a ** it also means that k is a “zero” (a solution) of the function!solution) of the function!

HW:

Factoring a polynomialFactoring a polynomial

Factor f(x) = 2xFactor f(x) = 2x33 + 11x + 11x22 + 18x + 9 + 18x + 9Given f(-3)=0Given f(-3)=0

*Since the remainder is *Since the remainder is zerozero, , x – k is a factorx – k is a factor!! x-(-3)x-(-3) or or x+3x+3*So *So use synthetic divisionuse synthetic division to find the others!! to find the others!!

Ex 6:Ex 6:

continued…

Do Now: Do Now: Use synthetic division to divide Use synthetic division to divide f(x)= xf(x)= x44 -6x -6x33 -40x +33 -40x +33 by by x - 7x - 7

xx33+ x+ x22 + 7x +9 + + 7x +9 + 96 96 x – 7x – 7

22 11 11 18 18 9 9

-3-3 2

-6

5

-15

3

-9

R: 0

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

**keep factoringkeep factoring (bustin ‘da ‘b’) gives you: (bustin ‘da ‘b’) gives you:

(x+3)(2x+3)(x+1)

Factor f(x)= 3xFactor f(x)= 3x33 + 13x + 13x22 + 2x -8 + 2x -8

Given f(-4)=0Given f(-4)=0

(x + 1)(3x – 2)(x + (x + 1)(3x – 2)(x + 4)4)

Ex 7:Ex 7:

Finding the zeros of a polynomial Finding the zeros of a polynomial functionfunction

One zero of f(x) = xOne zero of f(x) = x33 – 2x – 2x22 – 9x +18 is – 9x +18 is x=2. Find the others.x=2. Find the others.

*if *if x=2 x=2 is a zero, then is a zero, then x-2x-2 is a is a factorfactor

**Use synthetic divisionUse synthetic division to factor completely to factor completely..**then solvethen solve

(x-2)(x(x-2)(x22-9) -9) (x-2)(x+3)(x-3)(x-2)(x+3)(x-3)x=2 x=-3 x=3x=2 x=-3 x=3

Ex 8:Ex 8:

One zero of f(x) = xOne zero of f(x) = x33 + 6x + 6x22 + 3x -10 is + 3x -10 is x=-5. Find the others.x=-5. Find the others.

(x-2)(x+1)(x+5)(x-2)(x+1)(x+5)

x=2 x=-1 x=-5x=2 x=-1 x=-5

HW:

Ex 9:Ex 9: