Warm-Up Exercises

Find the product.

1. (x + 6)(x – 4)

2. (2y + 3)( y + 5)

ANSWER x2 + 2x – 24

ANSWER 2y2 + 13y + 15

Warm-Up Exercises

ANSWER 2x2 – 3x – 2; 18in.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?

Warm-Up ExercisesFactor when b and c are positiveEXAMPLE 1

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

18 + 1 = 19

9 + 2 = 11

6 + 3 = 9

Correct sum

Warm-Up ExercisesFactor when b and c are positiveEXAMPLE 1

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)

(x + 9)(x + 2) Multiply binomials.= x2 + 2x + 9x + 18

Simplify.

CHECK

= x2 + 11x + 18

Warm-Up ExercisesGUIDED PRACTICE for Example 1

Factor the trinomial

1. x2 + 3x + 2

ANSWER (x + 2)(x + 1)

2. a2 + 7a + 10

ANSWER (a + 5)(a + 2)

3. t2 + 9t + 14.

ANSWER (t + 7)(t + 2)

Warm-Up ExercisesFactor when b is negative and c is positiveEXAMPLE 2

Factor n2 – 6n + 8.

Because b is negative and c is positive, p and q must both be negative.

ANSWER

n2 – 6n + 8 = (n – 4)( n – 2)

Factors of 8 Sum of factors

–8, –1

–4, –2

–8 + (–1) = –9

–4 + (–2) = –6 Correct sum

Warm-Up ExercisesFactor when b is positive and c is negativeEXAMPLE 3

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

–15 + 1 = –14

15 + (–1) = 14

–5 + 3 = –2

15, –1

5, –3 5 + (–3) = 2 Correct sum

ANSWER y2 + 2y – 15 = (y + 5)( y – 3)

Warm-Up Exercises

4. x2 – 4x + 3.

ANSWER (x – 3)( x – 1)

GUIDED PRACTICE for Examples 2 and 3

Factor the trinomial

5. t2 – 8t + 12.

ANSWER (t – 6)( t – 2)

Warm-Up Exercises

6. m2 + m – 20.

GUIDED PRACTICE for Examples 2 and 3

Factor the trinomial

ANSWER (m + 5)( m – 4)

7. w2 + 6w – 16.

ANSWER (w + 8)( w – 2)

Warm-Up ExercisesEXAMPLE 4 Solve a polynomial equation

Solve the equation x2 + 3x = 18.

Write original equation.x2 + 3x = 18

Subtract 18 from each side.x2 + 3x – 18 = 0

Factor left side.(x + 6)(x – 3) = 0

Zero-product propertyx – 3 = 0x + 6 = 0

Solve for x.x = 3

or

orx = – 6

ANSWER The solutions of the equation are – 6 and 3.

Warm-Up ExercisesEXAMPLE 4 Solve a polynomial equation

8. Solve the equation s2 – 2s = 24.

ANSWER The solutions of the equation are – 4 and 6.

GUIDED PRACTICE for Example 4

Warm-Up ExercisesEXAMPLE 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.

STEP 1Draw a diagram of two banners together.

SOLUTION

Warm-Up ExercisesEXAMPLE 5 Solve a polynomial equation

STEP 2Write an equation using the fact that the area of 2 banners is 2(16.5) = 33 square feet. Solve the equation for w.

Formula for area of a rectangleA = l w

Substitute 33 for A and (4 + w + 4) for l.

Simplify and subtract 33 from each side.0 = w2 + 8w – 33

Factor right side.0 = (w + 11)(w – 3)

Zero-product propertyw + 11 = 0 or w – 3 = 0

33 = (4 + w + 4) w

Solve for w.w = – 11

ANSWERThe banner cannot have a negative width, so the width is 3 feet.

w = 3or

Warm-Up ExercisesGUIDED 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

Warm-Up ExercisesDaily Homework Quiz

Factor the trinomial.

1. x2 – 6x – 16

2. y2 + 11y + 24

ANSWER (x +2)(x – 8)

3. x2 + x – 12

ANSWER (y +3)(y + 8)

ANSWER (x +4)(x – 3)

Warm-Up ExercisesDaily Homework Quiz

4. Solve a2 – a = 20

ANSWER – 4, 5

Each wooden slat on a set of blinds has width w and length w + 17. The area of one slat is 38 square inches. What are the dimensions of a slat?

5.

ANSWER 2 in. by 19 in.