Download The Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra
Joseph Lee
Metropolitan Community College
Joseph Lee
The Fundamental Theorem of Algebra
The Factor Theorem
Let p(x) and d(x) be polynomials. If r (x) = 0 by the division
algorithm, then d(x) is a factor of p(x).
Joseph Lee
The Fundamental Theorem of Algebra
Example 1.
Use the remainder theorem to verify that −3 is a zero of
f (x) = x 3 − 3x 2 − 10x + 24. Then find all other zeros.
Joseph Lee
The Fundamental Theorem of Algebra
Example 2.
Use the remainder theorem to verify that 7 is a zero of
f (x) = x 3 − 5x 2 − 13x − 7. Then find all other zeros.
Joseph Lee
The Fundamental Theorem of Algebra
The Rational Zeros Theorem
Let p(x) be a polynomial function with integer coefficients:
p(x) = an x n + an−1 x n−1 + ... + a1 x + a0 .
Then any rational zero of the polynomial will be of the form
±
factor of a0
factor of an
leading coefficient an and constant term a0 .
Joseph Lee
The Fundamental Theorem of Algebra
Example 3.
List all possible rational zeros for f (x) = x 3 − 3x 2 − 10x + 24
given by the Rational Zeros Theorem.
Joseph Lee
The Fundamental Theorem of Algebra
Example 4.
List all possible rational zeros for f (x) = 2x 3 + 3x 2 − 32x + 15
given by the Rational Zeros Theorem.
Joseph Lee
The Fundamental Theorem of Algebra
Descartes’ Rule of Signs
Let p(x) be a polynomial function.
The number of positive real zeros is equal to or less than by an
even number the number of sign changes of p(x).
The number of negative real zeros is equal to or less than by an
even number the number of sign changes of p(−x).
Joseph Lee
The Fundamental Theorem of Algebra
The Fundamental Theorem of Algebra
Let p(x) be a polynomial function of degree n. Then p(x) has n
complex zeros, including multiplicities.
Joseph Lee
The Fundamental Theorem of Algebra
Example 5.
Find all zeros of the function f (x) = x 3 − 4x 2 + x + 6.
Joseph Lee
The Fundamental Theorem of Algebra
Example 6.
Find all zeros of the function f (x) = x 3 + 7x 2 + 16x + 12.
Joseph Lee
The Fundamental Theorem of Algebra
Example 7.
Find all zeros of the function f (x) = x 4 − 4x 3 − 19x 2 + 46x − 24.
Joseph Lee
The Fundamental Theorem of Algebra
Example 8.
Find all zeros of the function f (x) = x 4 − x 3 − 2x 2 − 4x − 24.
Joseph Lee
The Fundamental Theorem of Algebra
Complex Conjugate Theorem
Let p(x) be a polynomial with real coefficients. If a + bi is a zero
of the polynomial, then its complex conjugate a − bi is also a zero
of the polynomial.
Joseph Lee
The Fundamental Theorem of Algebra
Example 9.
Find a third degree polynomial f (x) with zeros of i and 3 such
that f (0) = −3.
Joseph Lee
The Fundamental Theorem of Algebra
Example 10.
Find a third degree polynomial f (x) with zeros of 1 + i and −1
such that f (1) = 2.
Joseph Lee
The Fundamental Theorem of Algebra