Avon High School
Section: 3.4
ACE COLLEGE ALGEBRA II - NOTES
Zeros of Polynomial Functions
Mr. Record: Room ALC-129
Day 1 of 1
STORY TIME We’ll start this section with an interesting and disturbing piece of Mathematics History.
No drawn portrait
ever found
Nicolo Tartaglia (1499-1557)
Girolamo Cardano (1501-1576)
Lodovico Ferrari (1522-1565)
Evariste Galois (1811-1832)
The Rational Zero Theorem
The Rational Zero Theorem
If f ( x)  an x n  an 1 x n 1  an  2 x n  2
 a1 x  a0 has integer coefficients and
p
q


p
is reduced to lowest terms  is a rational zero of f, then p is a factor of the constant term
 where
q


a0 , and q is a factor of the leading coefficient an .
In summary:
Possible rational zeros 
Example 1
Factors of the constant term
Factors of the leading coefficient
Using the Rational Root Theorem
List all possible rational zeros of f ( x)  4 x3  2 x 2  x  6 .
Example 2
Example 3
Example 4
Finding Zeros of a Polynomial Function
Find all zeros of f ( x)  x3  8x 2  11x  20 .
Finding Zeros of a Polynomial Function
Find all zeros of f ( x)  x3  x 2  5 x  2 .
Solving a Polynomial Equation
Solve x 4  6 x3  22 x 2  30 x  13  0
Properties of Roots of Polynomial Equations
1. If a polynomial equation is of degree n, then counting multiple roots separately, the solution has n
roots.
2. If a  bi is a root of a polynomial equation with real coefficients ( b  0 ), then the imaginary
number a  bi is also a root. Imaginary roots, if they exist, occur in conjugate pairs.
The Fundamanetal Theorem of Algebra
In 1799, a 22-year old student named Carl Friedrich Gauss discovered the following
theorem.
The Fundamental Theorem of Algebra
If f ( x) is a polynomial of degree n, where n  1, then the equation f ( x)  0
has at least one complex root.
Carl Friedrich Gauss (1777-1855)
The Linear Factorization Theorem
The Linear Factorization Theorem
If f ( x)  an x n  an 1 x n 1  an  2 x n  2
 a1 x  a0 , where n  1 and an  0 , then
f ( x)  an ( x  c1 )( x  c2 ) ( x  cn ),
where c1 , c2 , cn are complex numbers (possibly real and not necessarily distinct).
In other 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.
number
is also a root. Imaginary roots, if they exist, occur in conjugate pairs.
Example 5
Finding a Polynomial Function with Given Zeros
Find a fourth-degree polynomial function f ( x) with real coefficients that has
3, 2, and 1  i as roots.
Descartes’s Rule of Signs
Descartes’s Rule of Signs
Let f ( x)  an x n  an 1 x n 1  an  2 x n  2  a1 x  a0 be a polynomial with real coefficients.
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 the number of sign changes of f ( x) by a positive even integer.
If f ( x) has only one variation 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 the number of sign changes of f ( x) by a positive even integer.
If f ( x) has only one variation in sign, then f has exactly one negative real zero.
Example 6
Using Descartes’s Rule of Signs
Determine the possible numbers of positive and negative real zeros of
f ( x)  x 4  14 x3  71x 2  154 x  120 .