Download 4.3 Roots of Polynomial

Section 4.3
Roots of Polynomial Functions
Note: a 2 + b 2 can be factored over the complex numbers: a 2 + b 2 = (a + bi)(a – bi)
Factor 36 x 2 + 25
Example: Factor x 2 + 4
Example 1: Find the real and complex zeros and their multiplicities of the given
polynomial.
b. f ( x) = x 2 − 25
a. f ( x) = 4 x 2 + 9
c. f ( x) = x 2 − 7
d. f ( x) = x 3 − 11x
e. f ( x) = x 3 − 9 x 2 − 4 x + 36
f. f ( x) = x 3 − 4 x 2 + x − 4
g. f ( x) = ( x − 3) 5 ( x 2 − 3 x − 10)
Section 4.3 – Roots of Polynomials
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Properties of Polynomials
1. If a polynomial equation is of degree n, then counting multiple roots separately, the
equation has n roots.
2. If a + bi is a root of a polynomial equation (b is not equal 0), then the complex
number a – bi is also a root.
The Factor Theorem
Let f(x) be a polynomial.
a. If x – c is a factor of f(x), then c is a zero of f(x). (We saw this in Section 4.1.)
Example: Let (x – 7) be a factor of f. What is a zero of f ?
b. . If c is a zero of f(x), then x – c is a factor of f(x).
Example: Let 2 be a zero of f. Give a factor.
Let -10 be a zero of f. Give a factor.
The Linear Factorization Theorem
If f(x)= a n x n + ... + a1 x + a 0 , where n > 1 and a n ≠ 0 , then
f ( x) = a n ( x − c1 )( x − c 2 ) ⋅ ⋅ ⋅ ( x − c n ) ,
where c1 , c 2 ,..., c n 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.
Example 2: Find a 3rd degree polynomial with integer coefficients given that -7i and
1
2
are zeros.
Section 4.3 – Roots of Polynomials
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Example 3: Find a 3rd degree polynomial with integer coefficients given that -5 and i are
zeros and constant coefficient is -25.
Section 4.3 – Roots of Polynomials
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