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Remainder and Factor Theorems
Unit 11
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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.
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Remainder TheoremIf the polynomial expression that defines the function of P is divided by x-r, then the remainder is the number P(r).
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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.
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Integer Roots
Unit 11
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Page 464, #42-4442)
43)
44)
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Warm UpFind the polynomial P(x) in the standard
form that has roots of x = {-3, -1, 1} and P(0) = 9.
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QuizFactor & Remainder Theorem
.
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ExampleAs the first step in creating a graph of the polynomial, find all x-intercepts of the polynomial:
.
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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.
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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.
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Examples1.)
2.)
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Examples3.)
4.)
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Assignment
Worksheet #1, 1-5
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Show What You Know
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Rational Roots
Unit 11
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Warm UpList all possible roots and use them to
find the zeros of the polynomial:
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Worksheet #1, 1-51) -6, -1, 12) -3, -2, 23) -4, 2 (multiplicity 2)4) -1 (multiplicity 2), 25) -3, -1, 2, 3
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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.
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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.
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Examples1.)
2.)
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Assignment
Worksheet #2, 1-16
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Show What You Know
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Rational Roots
Unit 11
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Warm UpList all possible roots and use them to
find the zeros of the polynomial:
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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
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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}
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QuizInteger and Rational Roots
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Graphing Polynomials
Unit 11
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Warm UpDetermine the number of roots. Then find the
roots of the polynomial.
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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?
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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.
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End Behaviorf(x)=axn+…a > 0 a < 0
left right left rightn is even
n is odd
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Leading Coefficient > 0
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Leading Coefficient < 0
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End Behaviorf(x)=axn+…a > 0 a < 0
left right left rightn is even increase decrease
n is odd increase decrease
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Highest Exponent is Odd
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Highest Exponent is Even
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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
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ExamplesSketch the graph of each polynomial.1.)
2.)
3.)
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Assignment
Worksheet 3, #1-8
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Exit SurveyWhich of the following is the graph of
?
A.
B.
C. D.
B.
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Polynomial Review
Unit 11
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Warm Up Sketch a graph of the polynomial:
−2x3 + 2x2 +16x − 24
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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
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Determine if (x+1) is a factor of the polynomial:
A=YesB=No
2x3 −3x2 + x + 6
Example #2
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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)
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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
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Assignment
Review Sheet