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MATH 1100
SECTION 5.4 Notes
The Fundamental Theorem of Algebra – Text Pages 346-352
We have already seen that an nth degree polynomial can have at
most n real zeros. In the complex number system, an nth degree
polynomial has exactly n zeros, so it can be factored into exactly n linear
factors.
The Fundamental Theorem of Algebra:
Every polynomial Px   a n x n  a n 1 x n 1  ...  a1 x  a0
complex coefficients has at least one complex zero.
n  1,
an  0 with
Complete Factorization Theorem:
If P(x) is a polynomial of degree n>o, then there exist complex
numbers a, c1 , c2 , ... , cn (with a  0 ) such that
Px   ax  c1 x  c2     x  cn  .
Zeros Theorem:
Every polynomial of degree n  1 has exactly n zeros, provided that
a zero of multiplicity k is counted k times.
Conjugate Zeros Theorem:
If the polynomial P has real coefficients, and if the complex number
z is a zero of P, then its complex conjugate, z is also a zero P.
Linear and Quadratic Factors Theorem:
Every polynomial with real coefficients can be factored into a
product of linear and irreducible quadratic factors with real coefficients.
Example 1:
Factor the polynomial completely and find all its real and complex
zeros.
State the multiplicity of each zero.
Px   x 3  8x 2  9 x  28
Example 2:
Find a polynomial with integer coefficients that satisfies the
following:
(a.)
P has degree 3 and zeros 3, 2i, and –2i, with P2  32 .
(b.)
P has degree 3 and zeros –3, and 2+4i.
Example 3:
For the polynomial Px   x 4  8 x 2  9
(a.) Factor P into linear and irreducible quadratic factors with real
coefficients.
(b.)
Factor P completely into linear factors with complex coefficients.