Question

1. $3b^3 - 5b^2 + 2b$
2. $7x^2 - 32x - 60$
3. $30n^2b - 87nb + 30b$
4. $9r^2 - 5r - 10$
5. $9p^2r + 73pr + 70r$
6. $9x^2 + 7x - 56$
7. $4x^3 + 43x^2 + 30x$
8. $10m^2 + 89m - 9$

critical thinking questions:

1. for what values of $b$ is the expression factorable?

$x^2 + bx + 12$

1. name four values of $b$ which make the expression factorable:

$x^2 - 3x + b$

Explanation:

11) Step1: Factor out GCF

$3b^3 - 5b^2 + 2b = b(3b^2 - 5b + 2)$

11) Step2: Factor quadratic trinomial

$3b^2 - 5b + 2 = (3b - 2)(b - 1)$

12) Step1: Factor quadratic trinomial

$7x^2 - 32x - 60 = (7x + 10)(x - 6)$

13) Step1: Factor out GCF

$30n^2b - 87nb + 30b = 3b(10n^2 - 29n + 10)$

13) Step2: Factor quadratic trinomial

$10n^2 - 29n + 10 = (5n - 2)(2n - 5)$

14) Step1: Factor quadratic trinomial

$9r^2 - 5r - 10 = (9r + 10)(r - 1)$

15) Step1: Factor out GCF

$9p^2r + 73pr + 70r = r(9p^2 + 73p + 70)$

15) Step2: Factor quadratic trinomial

$9p^2 + 73p + 70 = (9p + 10)(p + 7)$

16) Step1: Factor quadratic trinomial

$9x^2 + 7x - 56 = (9x - 21)(x + 8/3) = (3x - 7)(3x + 8)$

17) Step1: Factor out GCF

$4x^3 + 43x^2 + 30x = x(4x^2 + 43x + 30)$

17) Step2: Factor quadratic trinomial

$4x^2 + 43x + 30 = (4x + 3)(x + 10)$

18) Step1: Factor quadratic trinomial

$10m^2 + 89m - 9 = (10m - 1)(m + 9)$

19) Step1: Identify factor pairs of 12

Factor pairs of 12: $(1,12), (2,6), (3,4), (-1,-12), (-2,-6), (-3,-4)$

19) Step2: Sum pairs for $b$

$b = 1+12=13;\ b=2+6=8;\ b=3+4=7;\ b=-1+(-12)=-13;\ b=-2+(-6)=-8;\ b=-3+(-4)=-7$

20) Step1: Identify factor pairs for $b$

For $x^2-3x+b$, pairs sum to 3: $(1,2), (0,3), (-1,4), (-2,5)$

20) Step2: Multiply pairs for $b$

$b=1×2=2;\ b=0×3=0;\ b=-1×4=-4;\ b=-2×5=-10$

Answer:

1. $b(3b - 2)(b - 1)$
2. $(7x + 10)(x - 6)$
3. $3b(5n - 2)(2n - 5)$
4. $(9r + 10)(r - 1)$
5. $r(9p + 10)(p + 7)$
6. $(3x - 7)(3x + 8)$
7. $x(4x + 3)(x + 10)$
8. $(10m - 1)(m + 9)$
9. $b = 13, 8, 7, -13, -8, -7$
10. $b = 2, 0, -4, -10$ (example values)