Remainder and Factor Theorems

Unit 11

DefinitionsThe real number, r, is a zero of f(x) iff:r is a solution, or root, of f(x)=0x-r is a factor of the expression that defines f (f(r)=0)when the expression is divided by x-r, the remainder is 0r is an x-intercept of the graph of f.

Remainder TheoremIf the polynomial expression that defines the function of P is divided by x-r, then the remainder is the number P(r).

Factor Theoremx-r is a factor of the polynomial expression that defines the function P iff r is a solution of P(x)=0. That is, if P(r)=0.

Integer Roots

Unit 11

Page 464, #42-4442)

43)

44)

Warm UpFind the polynomial P(x) in the standard

form that has roots of x = {-3, -1, 1} and P(0) = 9.

QuizFactor & Remainder Theorem

.

ExampleAs the first step in creating a graph of the polynomial, find all x-intercepts of the polynomial:

.

Rational Root Theorem

Let P be a polynomial function with integer coefficients in standard form. If is a root of P(x) = 0, then

p is a factor of the constant term of P and

q is a factor of the leading coefficient of P.

Determine the number of roots.

List all factors of the constant term.

List all factors of the leading coefficient.

List all the possible roots.

Test each possible root to find the zeros of each polynomial.

Examples1.)

2.)

Examples3.)

4.)

Assignment

Worksheet #1, 1-5

Show What You Know

Rational Roots

Unit 11

Warm UpList all possible roots and use them to

find the zeros of the polynomial:

Worksheet #1, 1-51) -6, -1, 12) -3, -2, 23) -4, 2 (multiplicity 2)4) -1 (multiplicity 2), 25) -3, -1, 2, 3

Rational Root Theorem

Let P be a polynomial function with integer coefficients in standard form. If is a root of P(x) = 0, then

p is a factor of the constant term of P and

q is a factor of the leading coefficient of P.

Determine the number of roots.

List all factors of the constant term.

List all factors of the leading coefficient.

List all the possible roots.

Test each possible root (using substitution or synthetic division) to find the zeros of each polynomial.

Examples1.)

2.)

Assignment

Worksheet #2, 1-16

Show What You Know

Rational Roots

Unit 11

Warm UpList all possible roots and use them to

find the zeros of the polynomial:

Worksheet #2, 1-161) 1, 1/3 2) 1, 2, 4, 8, 16, 32, 643) 1, 2, 5, 104) 1, 2, 4, 8, 1/5, 2/5, 4/5, 8/55) 1, 5, 25, ½, 5/2, 25/2, ¼, 5/4, 25/46) 1, 3, 7, 21, 1/5, 3/5, 7/5, 21/57) 1, 3, 9, 278) 1, 7, ½, 7/2

Worksheet #2, 1-169) x={1 (multiplicity 2), -3}10) x={1 (multiplicity 2), 11}11) x={-1 (multiplicity 2), -2}12) x={-1, 1/5, -5}13) x={1 (multiplicity 2), ¼}14) x={-1, 1/3, -3}15) x={1 (multiplicity 2), 1/5, 7}16) x={-1 (multiplicity 2), 1/3, 5}

QuizInteger and Rational Roots

Graphing Polynomials

Unit 11

Warm UpDetermine the number of roots. Then find the

roots of the polynomial.

Critical ThinkingIn the process of solving you test 1, 2, 5, and 10 as possible zeros and determine that none of them are actual zeros. You then discover that -5/2 is a zero. You calculate the depressed polynomial to be Do you need to test 1, 2, 5, and 10 again? Why or why not?

End BehaviorWhat happens to a

polynomial function as its x-values get very small and very large is called the end behavior of the function.

End Behaviorf(x)=axn+…a > 0 a < 0

left right left rightn is even

n is odd

Leading Coefficient > 0

Leading Coefficient < 0

End Behaviorf(x)=axn+…a > 0 a < 0

left right left rightn is even increase decrease

n is odd increase decrease

Highest Exponent is Odd

Highest Exponent is Even

End Behaviorf(x)=axn+…a > 0 a < 0

left right left rightn is even increase increase decrease decrease

n is odd decrease increase increase decrease

ExamplesSketch the graph of each polynomial.1.)

2.)

3.)

Assignment

Worksheet 3, #1-8

Exit SurveyWhich of the following is the graph of

?

A.

B.

C. D.

B.

Polynomial Review

Unit 11

Warm Up Sketch a graph of the polynomial:

−2x3 + 2x2 +16x − 24

A=True B=False

a) If f(-5)=0, then (x-5) is a factor of f(x).b) If x=9 is a root of f(x), then (x-9) is a

factor of f(x).c) If the polynomial f(x) is synthetically

divided by (x-4) and the remainder is 0, then f(4)=0.

Example #1

Determine if (x+1) is a factor of the polynomial:

A=YesB=No

2x3 −3x2 + x + 6

Example #2

Example #3Find the polynomial, in factored form, with the roots x={-2,2,4} and f(1)=18. ABCD

(x +2)(x−2)(x−4)

12(x−2)(x+ 2)(x+ 4)

2(x +2)(x−2)(x−4)

(x−2)(x+ 2)(x+ 4)

Example #4

•How many roots will the function have? •List all the possible rational roots.•Perform the synthetic division.•Write the polynomial in its factored form with each factor having only integer coefficients.•Write the roots of the polynomial.•Sketch the graph.

2x3 −13x2 +17x+12

Assignment

Review Sheet