Question

factor each polynomial by factoring out the greatest common factor.

1. $4x + 10$
2. $12x - 2$
3. $9x^2 - 81x$
4. $-10m - 15m^2$
5. $3n + 7n^2$
6. $5mn + 7ab$
7. $12x^4 + 6x^3 - 8x^2$
8. $14x^2y + 12xy^2$
9. $14a^2b^2 + 21ab^2$
10. $25y^4 + 30y^2 - 5y$
11. $x^2y - x$
12. $28x^3y^5 + 21x^2y^3$
13. $30m^2 - 10mn - 60mn^2$
14. $24a^5b^7c^2 + 16a^5b^4c^3 - 32a^4bc^3$

3

Explanation:

1. Problem 1: $4x+10$

Step1: Identify GCF of 4,10

GCF of 4 and 10 is 2.

Step2: Factor out GCF

$2(2x + 5)$

2. Problem 2: $12x-2$

Step1: Identify GCF of 12,2

GCF of 12 and 2 is 2.

Step2: Factor out GCF

$2(6x - 1)$

3. Problem 3: $9x^2-81x$

Step1: Identify GCF of 9,81; $x^2,x$

GCF is $9x$.

Step2: Factor out GCF

$9x(x - 9)$

4. Problem 4: $-10m-15m^2$

Step1: Identify GCF of 10,15; $m,m^2$

GCF is $-5m$ (or $5m$, factoring out negative for standard form).

Step2: Factor out GCF

$-5m(2 + 3m)$

5. Problem 5: $3n+7n^2$

Step1: Identify GCF of 3,7; $n,n^2$

GCF is $n$.

Step2: Factor out GCF

$n(3 + 7n)$

6. Problem 6: $5mn+7ab$

Step1: Check for common factors

No shared variables/constants between terms.

Step2: State polynomial is prime

$5mn + 7ab$ (cannot be factored further)

7. Problem 7: $12x^4+6x^3-8x^2$

Step1: Identify GCF of 12,6,8; $x^4,x^3,x^2$

GCF is $2x^2$.

Step2: Factor out GCF

$2x^2(6x^2 + 3x - 4)$

8. Problem 8: $14x^2y+12xy^2$

Step1: Identify GCF of 14,12; $x^2,x; y,y^2$

GCF is $2xy$.

Step2: Factor out GCF

$2xy(7x + 6y)$

9. Problem 9: $14a^2b^2+21ab^2$

Step1: Identify GCF of 14,21; $a^2,a; b^2,b^2$

GCF is $7ab^2$.

Step2: Factor out GCF

$7ab^2(2a + 3)$

10. Problem 10: $25y^4+30y^2-5y$

Step1: Identify GCF of 25,30,5; $y^4,y^2,y$

GCF is $5y$.

Step2: Factor out GCF

$5y(5y^3 + 6y - 1)$

11. Problem 11: $x^2y-x$

Step1: Identify GCF of $x^2,x; y,1$

GCF is $x$.

Step2: Factor out GCF

$x(xy - 1)$

12. Problem 12: $28x^3y^5+21x^2y^3$

Step1: Identify GCF of 28,21; $x^3,x^2; y^5,y^3$

GCF is $7x^2y^3$.

Step2: Factor out GCF

$7x^2y^3(4xy^2 + 3)$

13. Problem 13: $30m^2-10mn-60mn^2$

Step1: Identify GCF of 30,10,60; $m^2,m,m$

GCF is $10m$.

Step2: Factor out GCF

$10m(3m - n - 6n^2)$

14. Problem 14: $24a^5b^7c^2+16a^5b^4c^3-32a^4bc^3$

Step1: Identify GCF of 24,16,32; $a^5,a^5,a^4; b^7,b^4,b; c^2,c^3,c^3$

GCF is $8a^4bc^2$.

Step2: Factor out GCF

$8a^4bc^2(3ab^6 + 2ab^3c - 4c)$

Answer:

1. $2(2x + 5)$
2. $2(6x - 1)$
3. $9x(x - 9)$
4. $-5m(2 + 3m)$
5. $n(3 + 7n)$
6. $5mn + 7ab$ (prime polynomial)
7. $2x^2(6x^2 + 3x - 4)$
8. $2xy(7x + 6y)$
9. $7ab^2(2a + 3)$
10. $5y(5y^3 + 6y - 1)$
11. $x(xy - 1)$
12. $7x^2y^3(4xy^2 + 3)$
13. $10m(3m - n - 6n^2)$
14. $8a^4bc^2(3ab^6 + 2ab^3c - 4c)$