2.5: Zeros of Polynomial Functions (Day 2)
Goals
Use the Fundamental Theorem of Algebra to determine the number of
zeros of polynomial functions.
Find rational zeros of a polynomial function.
Find conjugate pairs of complex zeros.
Find zeros of polynomials by factoring.
Use Descarte’s Rule of Signs and the Upper and Lower Bound Rules.
Rules for finding rational zeros
• Complex zeros always occur in conjugate pairs
• Rational Zero Test finds possible zeros
• Descartes' Rule of signs will show where zeros are located
• Upper/Lower Bounds will eliminate unnecessary zeros
Descartes's Rule of Signs
• The number of Positive Real Zero’s of f is either equal to the number of
variations in sign of f(x) or less than that number by an even integer.
• The number of Negative Real Zero’s of f is either equal to the number
of variations in sign of f(-x) or less than that number by an even integer.
Example 1: Use Descartes's Rule to determine possible # of +/- zeros.
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Sign. Sign. Everywhere a sign.
Upper and Lower Bounds
• Upper Bound - Possible zero is +, last row of synthetic division is + or 0
• Lower Bound - Possible zero is -, last row alternates + & - (0 is wild)
Example 2: Upper and Lower Bounds
Test for:
X=2
x=4
x=-4
Example 3: Find all zeros of f(x) = X5 + X3 + 2X2 – 12X + 8
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Do this. Don't do that. Can't you read the sign?
To Find ALL Zeros of a Polynomial
#1. p/q
#2. Descarte’s
#3. Upper/Lower Bound
#4. Get to a squared leading coefficient
Examples: Find ALL zeros of each polynomial.
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Mo' Zeros, Mo' Problems
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2.5 (Day 2) Assignment
p.160 #51, 53, 57, 59, 63, 65, 69, 71, 83, 85, 91-94 all (14
questions)
New directions #91-94 find ALL zeros. (real and complex)
All questions should be done without a graph!!
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2.5 (Day 2) KEY p.160 #51, 53, 57, 59, 63, 65, 69, 71, 83, 85, 91-94 all
92. -3/2, 1/3, 3/2
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