1. Solving Polynomials

2. Warm Up
Find the zeros of the following polynomials. 1. f(x) = x(x – 1)(x + 5) 2. g(x) = x3(x – 1)2(x + 5)7 Find the polynomial function with the given zeros. 1. 0, 1, -5

3. Multiplicity
The multiplicity of a zero is the power of the factor that it came from.
Find the zeros and their multiplicities for the following.
1. f(x) = (3x + 2)4(x – 7)
2. g(x) = (x + 3)4(5 – 4x)5(2x + 1)
3. h(x) = (9x – 2)2(x – 4)3(x + 6)5

4. Try This
Find the polynomial (in factored form) with the following zeros and multiplicities.
-1 (multiplicity 2), 0 (multiplicity 3), 4
3 (multiplicity 3), -5
0, -7 (multiplicity 2), 6 (multiplicity 4)

5. Fundamental Theorem of Algebra
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has exactly n solutions provided each solution with multiplicity 2 is counted twice, each solution with multiplicity 3 is counted as 3 solutions and so on.

6. Try This
For each of the following state how many solutions the polynomial has.
f(x) = x4 + 3x2 – x + 7
g(x) = -3x2 – 4x5 + 5x4 – 3x + 1
h(x) = (x + 1) 2(x – 3)(x + 6) 3
f(x) = (2x – 1)(3x + 2)(x – 4) 7

7. Solving Polynomials
Solve x3 – x2 – 30x = 0 If given no other information you will have to start by factoring or looking at the graph.

8. Solving Polynomials
Solve x4 + x3 – 12x2 = 0 if (x + 4) is a factor Solve x3 – 8x2 + 22x – 20 = 0 if 2 is a root.