Download 3.4 - Zeros of Polynomial Functions.notebook

3.4 ­ Zeros of Polynomial Functions.notebook
3.4: Zeros of Polynomial Functions
October 24, 2016
Date: 10/24
Rational Zero (Root) Theorem: If has integer coefficients, then any rational zeros (roots) of f(x) must be of the form where:
p is a factor of the constant term, _____ q is a factor of the leading coefficient, ____
So, possible rational zeros = Ex 1: List all of the possible rational zeros of
a)
b) 1
3.4 ­ Zeros of Polynomial Functions.notebook
October 24, 2016
*These potential rational zeros can be tested using ____________ division to determine if they are actual roots (zeros) of the polynomial function.
Ex 2: Find all of the zeros 2
3.4 ­ Zeros of Polynomial Functions.notebook
October 24, 2016
*Once the polynomial function is reduced to a quadratic, the remaining zeros can be found by factoring, completing the square, or by using the quadratic formula
Ex 3: Find all of the zeros of 3
3.4 ­ Zeros of Polynomial Functions.notebook
October 24, 2016
Properties of Polynomial Equations:
1. If a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots
2. If
is a root of a polynomial equation with real coefficients
, then the imaginary number _______ is also a root. Imaginary roots, if they exist, occur in complex conjugate pairs. Ex 4: Solve 4
3.4 ­ Zeros of Polynomial Functions.notebook
October 24, 2016
Fundamental Theorem of Algebra: If f(x) is a plynomial of degree n, where
, then the equation f(x) = 0 has at least one complex root. Recall: Complex numbers include real and imaginary.
The Linear Factorization Theorem: If
where
and
, then f(x) = __________________________________, where are complex numbers (possibly real and not necessarily distinct).
*In words = An nth degree polynomial can be expressed as the product of a nonzero constant and n linear factors, where each linear factor has a leading coefficient of 1.
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3.4 ­ Zeros of Polynomial Functions.notebook
October 24, 2016
Ex 5: Find a third‐degree polynomial function f(x) with real coefficients that has ‐3 and i as zeros and such that f(1) = 8.
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3.4 ­ Zeros of Polynomial Functions.notebook
Descartes’s Rule of Signs: Let
polynomial with real coefficients:
October 24, 2016
be a 1. The number of positive real zeros of f is either:
a. The same as the number of sign changes of f(x) or
b. Less than the number of sign changes of f(x) by a positive even integer. If f(x) has only one change in sign, then f has exactly one positive real zero.
2. The number of negative real zeros of f is either:
a. The same as the number of sign changes of f(‐x) or
b. Less than the number of sign changes of f(‐x) by a positive even integer. If f(‐x) has only one change in sign, then f has exactly one negative real zero.
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3.4 ­ Zeros of Polynomial Functions.notebook
October 24, 2016
Ex 6: Determine the possible number of positive and negative real zeros (and imaginary zeros) of
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3.4 ­ Zeros of Polynomial Functions.notebook
October 24, 2016
Ex 7: a) Find all of the zeros of the function (Consider Descartes’s Rule of signs and the Rational Roots Theorem)
b) Sketch a complete graph of f(x)
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3.4 ­ Zeros of Polynomial Functions.notebook
October 24, 2016
Homework: pg. 361 #9, 12, 15, 24, 26, 30, 37, 42, 46, 53, 58
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