Question

factor the following polynomials based on the examples shown in class. you will end up with two binomials at the end of the factoring process. there may or may not be a common factor. show your work!!! 10. $20n^2 - 52n - 24$ 11. $10k^2 + 75k + 35$ 12. $12x^2 + 2x - 2$ 13. $15y^2 - 18y - 24$ 14. $32x^2 - 16x + 2$ 15. $12m^2 - 24m - 15$ 16. $6a^2 + 9a - 27$ 17. $16x^2 - 26x + 8$ 18. $20x^2 + 55x + 30$

Explanation:

10. Step1: Factor out GCF

$20n^2 - 52n - 24 = 4(5n^2 - 13n - 6)$

10. Step2: Factor quadratic trinomial

Find two numbers: $2$ and $-15$, since $2 \times (-15) = -30$ and $2 + (-15) = -13$.
$5n^2 -13n -6 = 5n^2 +2n -15n -6 = n(5n+2) -3(5n+2) = (5n+2)(n-3)$

10. Step3: Combine factors

$4(5n+2)(n-3)$

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11. Step1: Factor out GCF

$10k^2 +75k +35 = 5(2k^2 +15k +7)$

11. Step2: Factor quadratic trinomial

Find two numbers: $1$ and $14$, since $1 \times 14 =14$ and $1+14=15$.
$2k^2 +15k +7 = 2k^2 +14k +k +7 = 2k(k+7) +1(k+7) = (2k+1)(k+7)$

11. Step3: Combine factors

$5(2k+1)(k+7)$

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12. Step1: Factor out GCF

$12x^2 +2x -2 = 2(6x^2 +x -1)$

12. Step2: Factor quadratic trinomial

Find two numbers: $3$ and $-2$, since $3 \times (-2) = -6$ and $3 + (-2)=1$.
$6x^2 +x -1 =6x^2 +3x -2x -1 =3x(2x+1) -1(2x+1)=(3x-1)(2x+1)$

12. Step3: Combine factors

$2(3x-1)(2x+1)$

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13. Step1: Factor out GCF

$15y^2 -18y -24 =3(5y^2 -6y -8)$

13. Step2: Factor quadratic trinomial

Find two numbers: $4$ and $-10$, since $4 \times (-10)=-40$ and $4+(-10)=-6$.
$5y^2 -6y -8=5y^2 +4y -10y -8=y(5y+4)-2(5y+4)=(5y+4)(y-2)$

13. Step3: Combine factors

$3(5y+4)(y-2)$

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14. Step1: Factor out GCF

$32x^2 -16x +2 =2(16x^2 -8x +1)$

14. Step2: Factor perfect square trinomial

$16x^2 -8x +1=(4x-1)^2$ (since $(a-b)^2=a^2-2ab+b^2$, $a=4x$, $b=1$)

14. Step3: Combine factors

$2(4x-1)^2$

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15. Step1: Factor out GCF

$12m^2 -24m -15=3(4m^2 -8m -5)$

15. Step2: Factor quadratic trinomial

Find two numbers: $2$ and $-10$, since $2 \times (-10)=-20$ and $2+(-10)=-8$.
$4m^2 -8m -5=4m^2 +2m -10m -5=2m(2m+1)-5(2m+1)=(2m+1)(2m-5)$

15. Step3: Combine factors

$3(2m+1)(2m-5)$

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16. Step1: Factor out GCF

$6a^2 +9a -27=3(2a^2 +3a -9)$

16. Step2: Factor quadratic trinomial

Find two numbers: $6$ and $-3$, since $6 \times (-3)=-18$ and $6+(-3)=3$.
$2a^2 +3a -9=2a^2 +6a -3a -9=2a(a+3)-3(a+3)=(2a-3)(a+3)$

16. Step3: Combine factors

$3(2a-3)(a+3)$

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17. Step1: Factor out GCF

$16x^2 -24x +8=8(2x^2 -3x +1)$

17. Step2: Factor quadratic trinomial

Find two numbers: $-1$ and $-2$, since $(-1) \times (-2)=2$ and $(-1)+(-2)=-3$.
$2x^2 -3x +1=2x^2 -2x -x +1=2x(x-1)-1(x-1)=(2x-1)(x-1)$

17. Step3: Combine factors

$8(2x-1)(x-1)$

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18. Step1: Factor out GCF

$20x^2 +55x +30=5(4x^2 +11x +6)$

18. Step2: Factor quadratic trinomial

Find two numbers: $8$ and $3$, since $8 \times 3=24$ and $8+3=11$.
$4x^2 +11x +6=4x^2 +8x +3x +6=4x(x+2)+3(x+2)=(4x+3)(x+2)$

18. Step3: Combine factors

$5(4x+3)(x+2)$

Answer:

1. $4(5n+2)(n-3)$
2. $5(2k+1)(k+7)$
3. $2(3x-1)(2x+1)$
4. $3(5y+4)(y-2)$
5. $2(4x-1)^2$
6. $3(2m+1)(2m-5)$
7. $3(2a-3)(a+3)$
8. $8(2x-1)(x-1)$
9. $5(4x+3)(x+2)$