1. 6.7 Using the Fundamental Theorem of Algebra

2. The Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n where n > 0, then the equation f(x)=0 has at least one root in the set of complex numbers.

3. Finding the Number of Solutions or Zeros
State the number of solutions and tell what they are.
x2 – 14x + 49 = 0
x4 + 3x3 – 8x2 – 22x – 24 = 0
x3 – 3x + 52 = 0

4. Finding all the zeros of the polynomial function
Find all the zeros.
f(x) = x3 + x2 – x + 15
f(x) = x3 + 5x2 – 6
f(x) = x3 – x2 + 4x – 4

5. Finding all the zeros of the polynomial function
Find all the zeros.
f(x) = x3 + x2 + 9x + 9
f(x) = x4 + 5x3 + 10x2 + 20x + 24
f(x) = x4 – 6x2 + 5

6. Using Zeros to Write Polynomial Functions
Write a polynomial function of least degree that has real coefficients, a leading coefficient of 1 and 1, -2 + i, and -2 + i as zeros.
Write a polynomial function of least degree that has reals coefficients, a leading coefficient of 1, and 5, 2i, and -2i as zeros.

7. Approximate the Real Zeros
f(x) = x3 – 4x2 – 5x + 14
f(x) = x3 – 3x2 – 2x + 6

8. Story Problems
A rectangular piece of sheet metal is 10in wide. Squares of side length x are cut from the corners and the remaining piece is folded to make a box. The volume of the box is modeled by V(x) = 4x3 – 4x x. What size square can be cut from the corners to give a box with a volume of 25in3?