Section 2.5
Zeros of Polynomial Functions
Remember Our Polynomial Definition
Definition of a Polynomial Function
Let 𝑛 be a nonnegative integer and let 𝑎𝑛 , 𝑎𝑛−1 , . . . , 𝑎2 , 𝑎1 , 𝑎0 be real
numbers, with 𝑎𝑛 ≠ 0. The function defined by
𝑓 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + . . . + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0
is called a polynomial function of degree 𝒏. The number 𝑎𝑛 is
called the leading coefficient.
Characteristics of Polynomials
• They are smooth and continuous.
Concepts Covered Thus Far . . .
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Degree
Leading Coefficient
End Behavior
Zeros
Multiplicity of Zeros
Number of Zeros
Number of Extrema
Rational Root Theorem
• Used in factoring a polynomial to determine its
roots.
Rational Root Theorem
If 𝑓 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + . . . + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 has
𝑝
integer coefficients and (reduced) is a rational zero of 𝑓,
𝑞
then 𝑝 is a factor of the constant term, 𝑎0 , and 𝑞 is a factor
of the leading coefficient, 𝑎𝑛 .
Example 1
List all possible rational zeros.
a. 𝑓 𝑥 = 𝑥 3 + 3𝑥 2 − 6𝑥 − 8
b. 𝑓 𝑥 = 4𝑥 5 − 8𝑥 4 − 𝑥 + 2
The Fundamental Theorem of Algebra
• Proven in 1799 by 22 year old Carl Friedrich
Gauss in his dissertation.
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 1 complex root.
The Linear Factorization Theorem
Linear Factorization Theorem
If 𝑓 𝑥 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + . . . + 𝑎2 𝑥 2 + 𝑎1 𝑥 + 𝑎0 where
𝑛 ≥ 1, and 𝑎𝑛 ≠ 0, then
𝑓 𝑥 = 𝑎𝑛 𝑥 − 𝑐1 𝑥 − 𝑐2 ⋯ 𝑥 − 𝑐𝑛−1 𝑥 − 𝑐𝑛
where 𝑐1 , 𝑐1 , ⋯ 𝑐𝑛 are complex numbers (possibly real).
Properties of Roots
of Polynomial Equations
• A polynomial equation of degree 𝑛 has 𝑛
complex roots (counting multiplicity).
• Remember real numbers are complex numbers.
• If 𝑎 + 𝑏𝑖 𝑏 ≠ 0 , is a root, then so is its
conjugate 𝑎 − 𝑏𝑖.
• Hence, imaginary roots come in pairs.
Example 2
Find an nth-degree polynomial satisfying the given
conditions.
𝑛 = 3; 2 and 2𝑖 are zeros; 𝑓 1 = 10
Example 3
Find an nth-degree polynomial satisfying the given
conditions.
𝑛 = 3; −2 and 4 + 5𝑖 are zeros; 𝑓 1 = 102
Example 4
Find an nth-degree polynomial satisfying the given
conditions.
𝑛 = 4; 𝑖 and 3𝑖 are zeros; 𝑓 −1 = 60
Example 5
Solve the polynomial equation. Check graphically.
𝑥 4 − 4𝑥 3 − 57𝑥 2 − 62𝑥 − 10 = 0
Example 6
Solve the polynomial equation. Check graphically.
𝑥 4 − 4𝑥 3 − 31𝑥 2 − 32𝑥 − 6 = 0
Questions???
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