Download 2.5 The Fundamental Theorem of Algebra

2.5 The Fundamental
Theorem of Algebra
• The Fundamental Theorem of Algebra
– If f(x) is a polynomial of degree n, where n > 0,
then f has at least one zero in the complex
number system
• Linear Factorization Theorem
– If f(x) is a polynomial of degree n where n > 0, f
has precisely n linear factors
f(x) = an (x –c1)(x-c2)…(x-cn)
Where c1, c2,…cn are complex numbers
Find the zeros of the polynomial
f(x) = x3 + 4x
f(x) = x2 - 6x + 9
f(x) = x5 + x3 + 2x2 – 12x + 8
Complex Zeros Occur in Conjugate
Pairs
Let f(x) be a polynomial function that has real
coefficients. If a + bi, where b ≠ 0, is a zero of the
function, then the complex a – bi is also a zero
of the function.
Therefore: if 3 + 2i is a zero, then 3 – 2i is a zero
Writing a Polynomial with Given Zeros
Zeros: -1, 2, 3i
Zeros: 0, 3, 5, 2 – 5i
Finding the Zeros
Find all the zeros of f(x) = x4 -3x3 + 6x2 + 2x – 60
given that 1 + 3i is a zero
Factoring a Polynomial
• Every polynomial of degree n > 0 with real
coefficients can be written as the product of
linear and quadratic factors with real
coefficients, where the quadratic factor have
no real zeros.
• A quadratic with no real zeros is “irreducible
over the reals”
Factoring a Polynomial
Write the polynomial f(x) = x4 – x2 – 20
a) as the product of factors that are
irreducible over the rationals
b) as the product of factors that are
irreducible over the reals
c) in completely factored form
Homework
Page 140 #’s 27-49 ODD