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

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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

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**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

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**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.

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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)

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**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)

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**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)

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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)

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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)

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**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.

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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.

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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.

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**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.

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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

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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)

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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.

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