know how to factor a polynomial to find the zeros of the polynomial functions

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