Gifted Precalculus

1
Gifted Precalculus

• Section 2.5 The Fundamental Theorem of Algebra

2
The Fundamental Theorem of Algebra

• If f(x) is a polynomial of degree n, where ngt0, f
  has at least one zero in the complex number
  system.

3
Linear Factorization Theorem

• If f(x) is a polynomial of degree n where ngt0, f
  has precisely n linear factors
• f(x) an(x c1)(x c2) . . . (x cn)
• where c1, c2, . . . , cn are complex numbers

4
Complex Zeros Occur in Conjugate Pairs

• Let f(x) be a polynomial function that has real
  coefficients. If a bi, where b does not equal
  0, is a zero of the function, the conjugate a
  bi is also a zero.

5
Find all the zeros of the function and write the
polynomial as a product of linear factors.

• f(x) x2 6x 2
• h(x) x3 3x2 4x - 2

6
f(x) x4 8x3 17x2 8x 16

• A) Find all the zeros of the function.
• B) Write the polynomial as a product of linear
  factors.
• C) Use the factorization to determine the
  x-intercepts of the graph.
• D) Use a graphing calculator to verify that the
  real zeros are the only x-intercepts.

7
Find a polynomial function with integer
coefficients that has the given zeros.

• 6, -5 2i, -5 2i