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Warm-Up
 Factor the following using the given factor and
division.
 1. 𝑥 3 + 2𝑥 2 − 23𝑥 − 60; 𝑥 + 4
 2. 6𝑥 3 − 7𝑥 2 − 29𝑥 − 12; 3𝑥 + 4
 3. 𝑥 4 + 12𝑥 3 + 38𝑥 2 + 12𝑥 − 63; 𝑥 2 + 6𝑥 + 9
Zeros of Polynomial Functions
Real Zeros
 Recall that a polynomial function of degree n can have
at most n real zeros. These real zeros can be rational or
irrational.
 The Rational Zero Theorem describes how the
leading coefficient and constant term of a polynomial
function with integer coefficients can be used to
determine a list of all possible zeros.
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Rational Roots Theorem
 If f is a polynomial with leading coefficient a and y-intercept b,
𝑝
where 𝑏 ≠ 0 then every rational zero of f has the form 𝑞 , where
 p and q are relatively prime integers
 p is a factor of b
 q is a factor of the leading coefficient a
Once you know all of the possible rational zeros of a polynomial
function, you can use direct or synthetic substitution to determine
which, if any, are actual real zeros of the polynomial.
i was told to be rational
List all possible rational zeros of each function. Then
determine which, if any, are zeros.
 𝑓 𝑥 = 𝑥 3 + 2𝑥 + 1
 𝑓 𝑥 = 𝑥 4 + 4𝑥 3 − 12𝑥 − 9
 ℎ 𝑥 = 3𝑥 3 − 7𝑥 2 − 22𝑥 + 8
 𝑔 𝑥 = 2𝑥 3 − 4𝑥 2 + 18𝑥 − 36
 𝑓 𝑥 = 3𝑥 4 − 18𝑥 3 + 2𝑥 − 21
Fundamental Theorem of Algebra
 A polynomial function of degree n, where n>0, has
at least one zero (real or imaginary) in the complex
number system.
 Corollary: A polynomial function of degree n
has exactly n zeros, including repeated zeros, in
the complex number system.
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Complex number
 A complex number is in the form 𝑎 + 𝑏𝑖 where a and b
are real numbers and i is the imaginary unit where:
𝑖 2 = −1
or
𝑖 = −1
Linear Factorization Theorem
 If 𝑓(𝑥) is a polynomial function of degree n>0, then f
has exactly n linear factors and
𝑓 𝑥 = 𝑎𝑛 𝑥 − 𝑐1 𝑥 − 𝑐2 … (𝑥 − 𝑐𝑛 )
where 𝑎𝑛 is some nonzero real number and
𝑐1 , 𝑐2 , … 𝑐𝑛 are the complex zeros (including repeated
zeros) of f.
Conjugate Root Theorem
 According to the Conjugate Root Theorem, when a
polynomial equation in one variable with real
coefficients has a root of the form 𝑎 + 𝑏𝑖, where 𝑏 ≠ 0,
then its complex conjugate, 𝑎 − 𝑏𝑖, is also a root.
 You can use this theorem to write a polynomial
function given its complex zeros.
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Find a polynomial given its zeros
 Write a polynomial function of least degree with real
coefficients in standard form that has −2, 4, and 3 − 𝑖
as zeros.
Now you try…
Write a polynomial function of least degree with real
coefficients in standard form with the given zeros.
1. −3, 1(multiplicity:2), 4𝑖
2. 2 3, −2 𝑥, 1 + 𝑖
Irreducible over the reals
 A function has complex zeros when its factored form
contains a quadratic factor which is irreducible over
the reals. A quadratic expression is irreducible over
the reals when it has real coefficients but no real
zeros associated with it.
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Factoring polynomial functions
over the reals
 Every polynomial function of degree n>0 with real
coefficients can be written as the product of linear
factors and irreducible quadratic factors, each with
real coefficients.
Factor and find the Zeros of a
polynomial function
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 Consider 𝑘 𝑥 = 𝑥 − 18𝑥 + 30𝑥 − 19𝑥 + 30
a) Write 𝑘(𝑥) as the product of linear and irreducible
quadratic factors.
i.
List all possible real zeros
ii.
Graph the function and guess zeros
iii. Use synthetic division to prove they are zeros
(making sure to perform on depressed
polynomial when one works)
iv. Find all zeros until you get to an irreducible
factor.
b) Write 𝑘(𝑥) as a product of linear factors.
i.
Use quadratic equation
c) List all zeros of 𝑘(𝑥)
Practice!
1. 𝑓 𝑥 = 𝑥 4 + 𝑥 3 − 26𝑥 2 + 4𝑥 − 120
2. 𝑓 𝑥 = 𝑥 5 − 2𝑥 4 − 2𝑥 3 − 6𝑥 2 − 99𝑥 + 108
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Find all the zeros of a polynomial
when one is known
Find all complex zeros of:
𝑝 𝑥 = 𝑥 4 − 6𝑥 3 + 20𝑥 2 − 22𝑥 − 13
Given that 2 − 3𝑖 is a zero of 𝑝 𝑥 .
Fun! Practice! Yay!
1. 𝑔 𝑥 = 𝑥 4 − 10𝑥 3 + 35𝑥 2 − 46𝑥 + 10; 2 + 3
2. ℎ 𝑥 = 𝑥 4 − 8𝑥 3 + 26x 2 − 8x − 95; 1 − 6
 Through Rational Roots Theorem p127 #2-10 Even
 Through Complex Conjugates p127 #32-40 Even
 Through Irreducible Quadratics p127 #42-48 Even
 Through Given Complex Zeroes p127 #50-54 Even
 Other Problems: P128 #73, 76, 78, 80
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