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Analysis AB
U5D10
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Polynomial Functions
Theorems to Help Find Zeros
Fundamental Theorem of Algebra (& Linear Factorization Theorem)
In the complex number system consisting of all real and imaginary numbers, if P(x) is a
polynomial of degree n (n>0) with complex coefficients, then the equation P(x) = 0 has
exactly n roots (provided a double root is counted as 2 roots, a triple root is counted as 3
roots, and so on).
Descarte’s Rule of Signs
For a polynomial function F(x) with real coefficients and a non-zero constant term
1.The number of positive real zeros is either equal to the number of sign variations of f(x) or
less than that number by an even integer.
2.The number of negative real zeros is either equal to the number of sign variations in f(-x)
or less than that number by an even integer.
Rational Root Theorem
If a polynomial function f(x) has integer coefficients, the possible rational roots must be
p
of the form , where p is a factor of the constant term and q is a factor of the leading
q
coefficient.
Upper and Lower Bound Rule
Let f(x) be a polynomial function with real coefficients and a positive leading coefficient.
Suppose f(x) is divided by (x – c), using synthetic division.
1.If c>0 and each number in the result is either positive or zero, then c is an upper
bound for the real zeros of f.
2.If c<0 and the numbers in the result alternate positive and negative (zero counts as
positive or negative), then c is a lower bound for the real zeros of f.
Zeros in Conjugate Pairs
If f(x) is a polynomial function with real coefficients, then any complex zeros come in
pairs. Eg. If a + bi is a zero, then a – bi is also a zero.
If f(x) is a polynomial function with rational coefficients, and a and b are rational
numbers such that b is irrational. If a  b is a zero of f(x), then a  b is also a zero.
A useful little fact:
Given 2 roots the product of the factors can be found using
x2 – (sum of the roots)x + (product of the roots)
Ex. A:
Write a polynomial of degree 5 with rational coefficients and zeros of 5,  2 and 2i
Ex. B:
Write a polynomial of least degree with roots 5,
4
and 5  2i
3
How does the graph of each of the functions below relate to the zeros of the function.
1.
f(x) = x3 – 4x2 – 4x + 16
2.
g(x) = x4 – 3x2 – 4
Find all the zeros of the following polynomials.
a. Write the polynomial as the product of factors that are irreducible over the rationals.
b. Write the polynomial as the product of linear and quadratic factors that are irreducible
over the reals.
c. Write the polynomial as a product of linear factors. (Completely factored)
3.
f(x) = x4 – 81
4.
g(x) = x3 – 6x2 + 13x – 10