This document provides examples and explanations of factoring polynomials and finding zeros of polynomial functions. It covers factoring polynomials using the difference of squares, sum and difference of cubes, and perfect squares and cubes. It also shows how to find all zeros of polynomial functions by factoring them into linear and quadratic factors and setting each factor equal to zero.

1. Zeros of Polynomial
Functions

2. Give the prime factors of the following:
1. 28
2. 45
3. 54

3. Review:
1: One common factor. a x + a y = a (x + y)
2x + 2y = 2 (x + y)
2: Several grouped common
factor. a x + a y + b x + b y = a(x + y)
+ b(x + y) = (a + b ) (x + y)

4. 3. Difference of two squares
x 2 - y 2 = (x + y)(x - y)
x 2 - 49 = (x + 7)(x - 7)
4: Difference of two squares
(x + y) 2 - z 2 = (x + y + z)(x + y - z)
(2x + 3) 2 - 49 = (2x + 3 + 7)(2x + 3 - 7)

5. 5: Sum of two cubes.
x 3 + y 3 = (x + y)(x 2 - x y + y 2)
x3 + 27 = (x + 3)(x2- 3x + 9)
6: Difference of two cubes.
x 3 - y 3 = (x - y)(x 2 + x y + y 2)
x3 - 125 = (x - 5)(x2 + 5x + 25)

6. 7: Difference of fourth Powers.
x 4 - y 4 = (x 2 - y 2)(x 2 + y 2) =
(x + y)(x - y)(x 2 + y 2)
x 4 - 16 = (x 2 - 4)(x 2 + 4)
(x - 2)(x + 2)(x 2 + 4)

7. 8: Perfect square
x 2 + 2xy + y 2 = (x + y) 2
x 2 + 8x + 16 = (x + 4)
9: Perfect square
x 2 - 2xy + y 2 = (x - y) 2
x 2 - 10x + 25 = (x - 5)

8. 11: Perfect cube
x 3 - 3x 2y + 3xy 2 - y 3 = (x - y) 3
10: Perfect cube
x 3 + 3x 2y + 3xy 2 + y 3 = (x + y) 3
x 3 + 21x 2 + 147x + 343 = (x + 7) 3
x 3 - 6x 2 + 12x - 8 = (x - 2) 3

9. Factor the polynomials.
1: (x + 1) 2 - 4
2: x 2 + 5x + 6
3: x 3 - 1
4: x 3 - x 2 - 25x + 25

10. Find all zeros.
1) f (x) = (2x − 1)(x − 5)
2) f (x) = (x − 3)2(3x + 1)(x + 1)
3) f (x) = (2x + 1)(x + 1)3(x − 1)
4) f (x) = x(2x − 1)(x − 1)(x + 1)

11. 5) f (x) = x3 − 3x2 − 4x + 12
6) f (x) = x4 + 2x3 − 13x2 − 14x + 24
7) f (x) = x3 − 4x2 − 5x
8) f (x) = x3 − 2x2 + x
9) f(x) = x3 - 2x2 + 2x - 4
10) f(x) = x4 - 16x3 + 94x2 - 240x + 225