Check Your Skills Answer Key

1. x² + 13x + 36

   (x + 4)(x + 9)
   (x + 4)(x + 9)	F – multiply the first term in each parentheses: x × x = x2.
   (x + 4)(x + 9)	O – multiply the outer term in each: x × 9 = 9x.
   (x + 4)(x + 9)	I – multiply the inner term in each: 4 × x = 4x.
   (x + 4)(x + 9)	L – multiply the last term in each: 4 × 9 = 36.
   x2 + 9x + 4x + 36 → x2 + 13x + 36

2. y² − 3y − 18

   (y + 3)(y − 6)
   (y + 3)( y − 6)	F – multiply the first term in each parentheses: y × y = y2.
   (y + 3)(y − 6)	O – multiply the outer term in each: y × −6 = −6y.
   (y + 3)(y − 6)	I – multiply the inner term in each: 3 × y = 3y.
   (y + 3)(y − 6)	L – multiply the last term in each: 3 × −6 = −18.
   y2 − 6y + 3y − 18 → y2 − 3y − 18

3. x² + 10x + 21

   (x + 7)(3 + x)
   (x + 7)(3 + x)	F – multiply the first term in each parentheses: x × 3 = 3x.
   (x + 7)(3 + x)	O – multiply the outer term in each: x × x = x2.
   (x + 7)(3 + x)	I – multiply the inner term in each: 7 × 3 = 21.
   (x + 7)(3 + x)	L – multiply the last term in each: 7 × x = 7x.
   3x + x2 + 21 + 7x → x2 + 10x + 21

4. 4 + 8t
   4(1 + 2t)	Factor out a 4.

5. 5x + 25y
   5(x + 5y)	Factor out a 5.

6. 2x2 + 16x3
   2x2(1 + 8x)	Factor out a 2x2.

7. x = 2 OR 1

   (x − 2)(x − 1) = 0
   (x − 2) = 0 → x = 2	Remove the parentheses and solve for x.
   OR (x − 1) = 0 → x = 1	Remove the parentheses and solve for x.

8. x = −4 OR −5

   (x + 4)(x + 5) = 0
   (x + 4) = 0 → x = −4	Remove the parentheses and solve for x.
   OR (x + 5) = 0 → x = −5	Remove the parentheses and solve for x.

9. y = 3 OR −6

   (y − 3)(y + 6) = 0
   (y − 3) = 0 → y = 3	Remove the parentheses and solve for y.
   OR (y + 6) = 0 → y = −6	Remove the parentheses and solve for y.

10. (x + 3)(x + 11)

    x² + 14x + 33

    

    The numbers 1 and 33 and 3 and 11 multiply to 33, and the numbers 3 and 11 sum to 14 (x + 3)(x + 11)

11. (x − 5)(x − 9)

    x² − 14x + 45

    

    The numbers 1 and 45, 3 and 15, and 5 and 9 multiply to 45. The numbers 5 and 9 sum to 14. (x − 5)(x − 9)

12. (x + 6)(x − 3)

    x² + 3x − 18

    

    The numbers 1 and 18, 2 and 9, and 3 and 6 multiply to 18. The difference of 3 and 6 is 3. The middle term is positive, so the greater of the two numbers (6) is positive.

    (x + 6)(x − 3)

13. (x + 6)(x − 11)

    x² − 5x − 66

    

    The numbers 1 and 66, 2 and 33, 3 and 22, and 6 and 11 multiply to 66. The difference of 6 and 11 is 5.

    (x + 6)(x − 11)

14. x = 1 OR 2

    x² − 3x + 2 = 0

    

    The numbers 1 and 2 multiply to 2 and add to 3.

    (x − 1)(x − 2) = 0

15. x = 5 OR −7

    x² + 2x − 35 = 0

    

    The numbers 5 and 7 multiply to 35 and their difference is 2. The middle term is positive, so the greater of the two numbers, 7, is positive. Thus, (x − 5)(x + 7) = 0.

16. x = 2 OR 13

    x² − 15x = −26

    x² − 15x + 26 = 0    Add 26 to both sides so that the expression equals 0.

    

    The numbers 2 and 13 multiply to 26 and sum to 15.

    (x − 2)(x − 13) = 0

17. 2
    
    

18. x = 5
    

19. x = −1, 2; x ≠ 4
    

    The numerator is 0 if either (x + 1) or (x − 2) is 0. Thus, x = −1 or x = 2. However, x ≠ 4 because x = 4 would make the fraction undefined.

20. (2a + b)² = 0

    4a² + 4ab + b² = 0 → (2a)² + 2(2a)(b) + b² = 0 → (2a + b)(2a + b) = 0

21. (x + 11y)² = 0

    x² + 22xy + 121y² = 0 → x² + 2x(11y) + (11y)² = 0 → (x + 11y)(x + 11y) = 0