1. Find the product.
1. (x + 6)(x – 4)
ANSWER
x2 + 2x – 24
2. (2y + 3)( y + 5)
ANSWER
2y2 + 13y + 15

2. Find the product.
3. The dimensions of a rectangular print can be represented by x – 2 and 2x + 1. Write an expression that models the area of the print. What is its area if x is 4 inches?
ANSWER
2x2 – 3x – 2; 18in.2

3. Factor when b and c are positive
EXAMPLE 1
Factor when b and c are positive
Factor x2 + 11x + 18.
SOLUTION
Find two positive factors of 18 whose sum is 11.
Make an organized list.
Factors of 18
Sum of factors
18, 1
9, 2
6, 3
= 19
9 + 2 = 11
6 + 3 = 9
Correct sum

4. Factor when b and c are positive
EXAMPLE 1
Factor when b and c are positive
The factors 9 and 2 have a sum of 11, so they are the correct values of p and q.
ANSWER
x2 + 11x + 18 = (x + 9)(x + 2)
CHECK
(x + 9)(x + 2)
= x2 + 2x + 9x + 18
Multiply binomials.
= x2 + 11x + 18
Simplify.

5. GUIDED PRACTICE
for Example 1
Factor the trinomial
x2 + 3x + 2
ANSWER
(x + 2)(x + 1)
a2 + 7a + 10
ANSWER
(a + 5)(a + 2)
t2 + 9t + 14.
ANSWER
(t + 7)(t + 2)

6. Factor when b is negative and c is positive
EXAMPLE 2
Factor when b is negative and c is positive
Factor n2 – 6n + 8.
Because b is negative and c is positive, p and q must both be negative.
Factors of 8
Sum of factors
–8, –1
–4, –2
–8 + (–1) = –9
–4 + (–2) = –6
Correct sum
ANSWER
n2 – 6n + 8 = (n – 4)( n – 2)

7. Factor when b is positive and c is negative
EXAMPLE 3
Factor when b is positive and c is negative
Factor y2 + 2y – 15.
Because c is negative, p and q must have different signs.
Factors of –15
Sum of factors
–15, 1
–5, 3
– = –14
15 + (–1) = 14
–5 + 3 = –2
15, –1
5, –3
5 + (–3) = 2
Correct sum
ANSWER
y2 + 2y – 15 = (y + 5)( y – 3)

8. GUIDED PRACTICE
for Examples 2 and 3
Factor the trinomial
4. x2 – 4x + 3.
ANSWER
(x – 3)( x – 1)
5. t2 – 8t + 12.
ANSWER
(t – 6)( t – 2)

9. GUIDED PRACTICE
for Examples 2 and 3
Factor the trinomial
6. m2 + m – 20.
ANSWER
(m + 5)( m – 4)
7. w2 + 6w – 16.
ANSWER
(w + 8)( w – 2)

10. Solve a polynomial equation
EXAMPLE 4
Solve a polynomial equation
Solve the equation x2 + 3x = 18.
x2 + 3x = 18
Write original equation.
x2 + 3x – 18 = 0
Subtract 18 from each side.
(x + 6)(x – 3) = 0
Factor left side.
x + 6 = 0 or
x – 3 = 0
Zero-product property
x = – 6 or
x = 3
Solve for x.
ANSWER
The solutions of the equation are – 6 and 3.

11. EXAMPLE 4
GUIDED PRACTICE
Solve a polynomial equation
for Example 4
8. Solve the equation s2 – 2s = 24.
ANSWER
The solutions of the equation are – 4 and 6.

12. EXAMPLE 5
Solve a multi-step problem
BANNER DIMENSIONS
You are making banners to hang during school spirit week. Each banner requires 16.5 square feet of felt and will be cut as shown. Find the width of one banner.
SOLUTION
STEP 1
Draw a diagram of two banners together.

13. Solve a polynomial equation
EXAMPLE 5
Solve a polynomial equation
STEP 2
Write an equation using the fact that the area of 2 banners is 2(16.5) = 33 square feet. Solve the equation for w.
A = l w
Formula for area of a rectangle
33 = (4 + w + 4) w
Substitute 33 for A and (4 + w + 4) for l.
0 = w2 + 8w – 33
Simplify and subtract 33 from each side.
0 = (w + 11)(w – 3)
Factor right side.
w + 11 = 0 or
w – 3 = 0
Zero-product property
w = – 11 or
w = 3
Solve for w.
ANSWER
The banner cannot have a negative width, so the width is 3 feet.

14. GUIDED PRACTICE
for Example 5
WHAT IF? In example 5, suppose the area of a banner is to be 10 square feet. What is the width of one banner? 9.
ANSWER
2 feet

15. Daily Homework Quiz
Factor the trinomial.
1. x2 – 6x – 16
ANSWER
(x +2)(x – 8)
2. y2 + 11y + 24
ANSWER
(y +3)(y + 8)
3. x2 + x – 12
ANSWER
(x +4)(x – 3)

16. Daily Homework Quiz
4. Solve a2 – a = 20
ANSWER
– 4, 5
Each wooden slat on a set of blinds has width w and length w The area of one slat is 38 square inches. What are the dimensions of a slat? 5.
ANSWER
2 in. by 19 in.