2.8 Warm Up2.8 Warm UpFactor.1. 8x² + 26x + 15

2. 2x² + 15x + 7

3. 5x² - 18x - 8

2.8 Factor 2.8 Factor Special Special

ProductsProducts

EXAMPLE 1 Factor the difference of two squares

Factor the polynomial.

a. y2 – 16 =

b. 25m2 – 36

c. x2 – 49y2

EXAMPLE 2 Factor the difference of two squares

Factor the polynomial 8 – 18n2.

8 – 18n2 = 2(4 – 9n2)

= 2(2 + 3n)(2 – 3n)

Factor out common factor.

Difference of two squares pattern

GUIDED PRACTICE for Examples 1 and 2

Factor the polynomial.

1. 4y2 – 64 = (2y + 8)(2y – 8)

EXAMPLE 3 Factor perfect square trinomials

Factor the polynomial.

a.

n2 – 12n + 36

b. 9x2 – 12x + 4

c. 4s2 + 4st + t2

EXAMPLE 4 Factor a perfect square trinomial

Factor the polynomial –3y2 + 36y – 108.

–3y2 + 36y – 108 Factor out –3.

= –3(y – 6)2 Perfect square trinomial pattern

= –3(y2 – 12y + 36)

GUIDED PRACTICE for Examples 3 and 4

Factor the polynomial.

= (h + 2)22. h2 + 4h + 4

3. 2y2 – 20y + 50 = 2(y – 5)2

4. 3x2 + 6xy + 3y2= 3(x + y)2

EXAMPLE 5 Solve a polynomial equation

Solve the equation x2 + + = 0.23

x 19

Write original equation.

9x2 + 6x + 1 = 0 Multiply each side by 9.

(3x + 1)2 = 0

x = – 13

Solve for x.

x2 + + = 019

23

x

ANSWER

The solution of the equation is – .13

GUIDED PRACTICE for Examples 5 and 6

Solve the equation

5. a2 + 6a + 9 = 0 a = –3

6. w2 – 14w + 49 = 0 w = 7

7. n2 – 81= 0 n = – 9 or n = 9

FALLING OBJECT

SOLUTION

EXAMPLE 6 Solve a vertical motion problem

A window washer drops a wet sponge from a height of 64 feet. After how many seconds does the sponge land on the ground?

Use the vertical motion model to write an equation for the height h (in feet) of the sponge as a function of the time t (in seconds) after it is dropped.

EXAMPLE 6 Solve a vertical motion problem

The sponge was dropped, so it has no initial vertical velocity. Find the value of t for which the height is 0.

h = –16t2 + vt + s Vertical motion model

0 = –16t2 + (0)t + 64 Substitute 0 for h, 0 for v, and 64 for s.

0 = –16(t2 – 4) Factor out –16.

0 = –16(t – 2)(t +2) Difference of two squares pattern

t – 2 = 0 or t + 2 = 0 Zero-product property

t = 2 or t = –2 Solve for t.

Disregard the negative solution of the equation.

ANSWERThe sponge lands on the ground 2 seconds after it is dropped.