2.4 Real Zeros of Polynomial Functions
Name: ____________
Objective: Students will be able to divide polynomials using
synthetic and long division, apply the Remainder Theorem, Factor
Theorem and Rational Zeros Theorem and find upper and lower
bounds for zeros of polynomials.
Division Algorithm for Polynomials
Let f(x) and d(x) be polynomials with the degree of f greater than
or equal to the degree of d, and d(x)≠0. Then there are unique
polynomials q(x) and r(x), called the quotient and remainder, such
that
f(x) = d(x)q(x) + r(x),
where either r(x) = 0 or the degree of r is less than the degree of
d.
We'll typically write it in fraction form:
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Examples Divide using long division.
1.) (3x3 + 5x2 + 8x + 7)÷(3x + 2)
2.) (2x4 - x3 - 2)÷(2x2 + x + 1)
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-Synthetic Division: Alternative to long division. Use when dividing by a factor
of the form x - k.
Examples: Divide using synthetic division.
1.) (x2 + 20x + 91) ÷ (x + 7)
2.) (3x4 - 2x3 + 5x2 - 4x - 2) ÷(x + 1)
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Remainder Theorem If a polynomial f(x) is divided by x - k, then
the remainder is r = f(k).
Example Use the Remainder Theorem to find the remainder
when f(x) is divided by x - k.
1.) f(x) = x4 - 5; k = 1
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Factor Theorem A polynomial function f(x) has a factor x - k if
and only if f(k) = 0.
Examples Use the Factor Theorem to determine whether the
first polynomial is a factor of the second polynomial.
1.) x - 3; x3 - x2 - x - 15
2.) x + 1; 2x10 - x9 + x8 + x7 + 2x6 - 3
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Examples of rational zeros:
Examples of irrational zeros:
Rational Zeros Theorem:
Let f(x) = anxn + an-1xn-1 + ... + a1x + a0 be a polynomial.
Roots of f(x) are of the form p , where p is a factor
q
of a0 and q is a factor of an.
Example 1.) List the possible rational zeros of f(x) = 3x3 - 5x2 - 4x + 10.
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2.) Use the Rational Zeros Theorem to list all the possible rational
zeros of f(x) = 2x4 - 7x3 - 8x2 + 14x + 8. Find the rational zeros.
Then, factor completely over the real numbers.
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Upper and Lower Bound Tests for Real Zeros
Let f be a polynomial function of degree n≥1 with a positive
leading coefficient. Suppose f(x) is divided by x - k using
synthetic division.
-If k≥0 and every number in the last line is nonnegative (positive
or zero), then k is an upper bound for the real zeros of f.
-If k≤0 and the numbers in the last line are alternately
nonnegative and nonpositive, then k is a lower bound for the real
zeros of f.
Prove that all of the real zeros of f(x) = 2x4 - 7x3 - 8x2 + 14x + 8
must lie in the interval [-2,5].
Assignment: Pages 223-225: SHOW ALL WORK!!!!! 5, 7, 11, 17, 19, 23, 25, 29, 33, 35, 37, 41, 47, 51, 73
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