Mrs. McConaughy Honors Algebra 2

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Lesson 6.5: Finding Rational Zeros

Objective: To find the rational zeros of a polynomial function

Mrs. McConaughy Honors Algebra 2

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Before we start, recall:

A rational number is one that can be written ________________________

______________________________.

The real-number zeros of a polynomial function are either ________ or __________.

as the quotient (ratio) of two integers.

rational

irrational

Mrs. McConaughy Honors Algebra 2

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EXAMPLES Find the zeros of the following functions:

f(x) = x2 - 3x – 4 f(x) = x2 - 2

Mrs. McConaughy Honors Algebra 2

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The Rational Zero TestConsider the polynomial

f(x) = anxn + an-1x

n-1 + ... + a1x + a0

with integer coefficients. Every rational zero of f has the form

p = ______________________q

factors of constant termfactors of leading coefficient

Mrs. McConaughy Honors Algebra 2

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EXAMPLE 1

Find the rational zeros of the following function:

f(x) = x4- x3 – 5x2 + 3x + 6

STEP 1: Find all possible rational zeros: -------------

STEP 2: Use the Remainder Theorem and Synthetic Division to determine actual zeros:f(x) = ________________ ________________________________

±1, ±2, ±3, ±6±1

(x-1) (x-2)(x2-3)

Mrs. McConaughy Honors Algebra 2

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From the previous factorization, you can see that f has only ______

rational zeros: ______________ (The other two

zeros, ______, are irrational.)

two

± 3

f(x) = x4-x3–5x2+3x+6

NOTE: Graphing can speed up the process of finding the first zero.

x = – 1 and x = 2.

Mrs. McConaughy Honors Algebra 2

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EXAMPLE 2

Find the real zeros of the following function:

f(x) = 10x3 – 15x2 - 16x + 12STEP 1: Find all possible rational zeros: ------------

STEP 2: Use the Remainder Theorem and Synthetic Division to determine actual zeros. Shorten your search for the initial rational zero by graphing on a graphics calculator, first. f(x) = ___________________________

Mrs. McConaughy Honors Algebra 2

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The Fundamental Theorem of Algebra

Counting complex and repeated solutions, an nth-degree polynomial equation has exactly n solutions.

Carl Friedrich Gauss

Mrs. McConaughy Honors Algebra 2

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FINAL CHECKS FOR UNDERSTANDING

1. List the possible rational zeros given by the Rational-Zero Test for:

f(x) = 3x3 – 6x2 + 7x - 22. Factor x3 – 3x2 - 6x + 8 completely.3. Factor x3 +4x2 - x - 4 completely.4. Which is the more efficient method,

factoring or rational-zero test, for finding all real zeros of the function, g(x) = x3 + x2 -2x – 2? Explain.

Mrs. McConaughy Honors Algebra 2

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In 4-6, find all real zeros of the polynomial function.

y = 4x3 – 12x2 - x + 15

y = -4x3 + 15x2 - 8x - 3

y = -3x3 + 20x2 - 36x + 16

HOMEWORK ASSIGNMENT:

______________________________

Mrs. McConaughy Honors Algebra 2

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Three reasonable choices for x would be: _____________-6/5, 1/2, 2

f(x) = _________________________