factoring by greatest common factor
factor the greatest common factor out of the polynomial.
1. $8x^2 + 10x$ 2. $12y - 16$ 3. $-15d^4 + 45d^3$
4. $13a + 20b$ 5. $c^3 + c^2 - c$ 6. $6n^2 - 30n + 42$
7. $-7m^2 - 10m + 17$ 8. $18p^3 - 63p^2 - 9p$ 9. $18x^2 - 50y^2$
10. $100z^9 + 50z^6 - 75z^5$ 11. $36rs^4 - 108r^3s^3$ 12. $36k - 30$
13. $a^7b - a^{10}$ 14. $2c^3d^4 - 3c^4 + 4c^3$ 15. $3g^4 + 3g^7$
16. $18x^5 - 48x^4 + 56x^3 - 86x$ 17. $23y^{10} - 46y^7 + 68y^4 + 10y$

Question

factoring by greatest common factor
factor the greatest common factor out of the polynomial.
1. $8x^2 + 10x$  2. $12y - 16$  3. $-15d^4 + 45d^3$
4. $13a + 20b$  5. $c^3 + c^2 - c$  6. $6n^2 - 30n + 42$
7. $-7m^2 - 10m + 17$  8. $18p^3 - 63p^2 - 9p$  9. $18x^2 - 50y^2$
10. $100z^9 + 50z^6 - 75z^5$  11. $36rs^4 - 108r^3s^3$  12. $36k - 30$
13. $a^7b - a^{10}$  14. $2c^3d^4 - 3c^4 + 4c^3$  15. $3g^4 + 3g^7$
16. $18x^5 - 48x^4 + 56x^3 - 86x$  17. $23y^{10} - 46y^7 + 68y^4 + 10y$

Accepted Answer

Let's solve problem 1: $8x^2 + 10x$
# Explanation:
## Step 1: Find GCF of coefficients and variables
- Coefficients: GCF of 8 and 10 is 2.
- Variables: GCF of $x^2$ and $x$ is $x$.
- So GCF is $2x$.
## Step 2: Factor out GCF
Divide each term by $2x$:
$8x^2\div2x = 4x$, $10x\div2x = 5$
So $8x^2 + 10x = 2x(4x + 5)$
# Answer:
$2x(4x + 5)$

Let's solve problem 2: $12y - 16$
# Explanation:
## Step 1: Find GCF of 12 and 16
GCF of 12 and 16 is 4.
## Step 2: Factor out 4
Divide each term by 4: $12y\div4 = 3y$, $-16\div4 = -4$
So $12y - 16 = 4(3y - 4)$
# Answer:
$4(3y - 4)$

Let's solve problem 3: $-15d^4 + 45d^3$
# Explanation:
## Step 1: Find GCF of coefficients and variables
- Coefficients: GCF of 15 and 45 is 15. Consider the sign, we can factor out -15 or 15. Let's factor out -15.
- Variables: GCF of $d^4$ and $d^3$ is $d^3$.
- So GCF is $-15d^3$ (or $15d^3$, but factoring out -15d^3 will make the first term positive inside the parentheses).
## Step 2: Factor out $-15d^3$
$-15d^4\div(-15d^3)=d$, $45d^3\div(-15d^3)= - 3$
So $-15d^4 + 45d^3=-15d^3(d - 3)$ (or $15d^3(-d + 3)$)
# Answer:
$-15d^3(d - 3)$ (or $15d^3(3 - d)$)

Let's solve problem 4: $13a + 20b$
# Explanation:
## Step 1: Check GCF of 13 and 20
13 is a prime number. GCF of 13 and 20 is 1.
## Step 2: Since GCF is 1, the polynomial is already factored (or we can write it as $1(13a + 20b)$)
# Answer:
$13a + 20b$ (or $1(13a + 20b)$)

Let's solve problem 5: $c^3 + c^2 - c$
# Explanation:
## Step 1: Find GCF of terms
GCF of $c^3$, $c^2$ and $c$ is $c$.
## Step 2: Factor out $c$
$c^3\div c = c^2$, $c^2\div c = c$, $-c\div c=-1$
So $c^3 + c^2 - c = c(c^2 + c - 1)$
# Answer:
$c(c^2 + c - 1)$

Let's solve problem 6: $6n^2 - 30n + 42$
# Explanation:
## Step 1: Find GCF of 6, 30 and 42
GCF of 6, 30 and 42 is 6.
## Step 2: Factor out 6
$6n^2\div6 = n^2$, $-30n\div6=-5n$, $42\div6 = 7$
So $6n^2 - 30n + 42 = 6(n^2 - 5n + 7)$
# Answer:
$6(n^2 - 5n + 7)$

Let's solve problem 7: $-7m^2 - 10m + 17$
# Explanation:
## Step 1: Check GCF of -7, -10 and 17
7 is prime, GCF of 7, 10 and 17 is 1.
## Step 2: Since GCF is 1, the polynomial is already factored (or we can write it as $1(-7m^2 - 10m + 17)$ or $-1(7m^2 + 10m - 17)$)
# Answer:
$-7m^2 - 10m + 17$ (or $-1(7m^2 + 10m - 17)$)

Let's solve problem 8: $18p^3 - 63p^2 - 9p$
# Explanation:
## Step 1: Find GCF of 18, 63 and 9
GCF of 18, 63 and 9 is 9.
## Step 2: Factor out 9
$18p^3\div9 = 2p^3$, $-63p^2\div9=-7p^2$, $-9p\div9=-p$
Wait, actually, let's check the variables. GCF of $p^3$, $p^2$ and $p$ is $p$. So overall GCF is $9p$? Wait, 18, 63, 9: GCF of 18, 63, 9. 18 = 2×9, 63=7×9, 9=1×9. So GCF is 9. And variables: $p^3$, $p^2$, $p$ have GCF $p$. So overall GCF is $9p$? Wait no, 18p³, -63p², -9p. Let's factor out 9p:

$18p^3\div9p = 2p^2$, $-63p^2\div9p=-7p$, $-9p\div9p = - 1$

Wait, let's recalculate GCF of coefficients: 18, 63, 9. The GCF of 18, 63, 9 is 9. GCF of variables: $p^3$, $p^2$, $p$ is $p$. So GCF is $9p$.

## Step 2: Factor out $9p$
$18p^3 - 63p^2 - 9p=9p(2p^2 - 7p - 1)$
Wait, but let's check: 9p×2p²=18p³, 9p×(-7p)= -63p², 9p×(-1)= -9p. Correct.
Wait, but initially I thought GCF of coefficients is 9, and variables is p, so GCF is 9p. Yes.
# Answer:
$9p(2p^2 - 7p - 1)$

Let's solve problem 9: $18x^2 - 50y^2$
# Explanation:
## Step 1: Find GCF of 18 and 50
GCF of 18 and 50 is 2.
## Step 2: Factor out 2
$18x^2\div2 = 9x^2$, $-50y^2\div2=-25y^2$
So $18x^2 - 50y^2 = 2(9x^2 - 25y^2)$
Notice that $9x^2 - 25y^2$ is a difference of squares: $9x^2=(3x)^2$, $25y^2=(5y)^2$, so $9x^2 - 25y^2=(3x - 5y)(3x + 5y)$
So we can further factor: $2(3x - 5y)(3x + 5y)$
# Answer:
$2(3x - 5y)(3x + 5y)$ (or $2(9x^2 - 25y^2)$)

Let's solve problem 10: $100z^9 + 50z^6 - 75z^5$
# Explanation:
## Step 1: Find GCF of 100, 50 and 75
GCF of 100, 50 and 75 is 25.
## Step 2: Find GCF of variables $z^9$, $z^6$, $z^5$
GCF is $z^5$.
## Step 3: Factor out $25z^5$
$100z^9\div25z^5 = 4z^4$, $50z^6\div25z^5 = 2z$, $-75z^5\div25z^5=-3$
So $100z^9 + 50z^6 - 75z^5 = 25z^5(4z^4 + 2z - 3)$
# Answer:
$25z^5(4z^4 + 2z - 3)$

Let's solve problem 11: $36rs^4 - 108r^3s^3$
# Explanation:
## Step 1: Find GCF of 36 and 108
GCF of 36 and 108 is 36.
## Step 2: Find GCF of variables $rs^4$ and $r^3s^3$
GCF of $r$ and $r^3$ is $r$, GCF of $s^4$ and $s^3$ is $s^3$. So GCF of variables is $rs^3$.
## Step 3: Factor out $36rs^3$
$36rs^4\div36rs^3 = s$, $-108r^3s^3\div36rs^3=-3r^2$
So $36rs^4 - 108r^3s^3 = 36rs^3(s - 3r^2)$
# Answer:
$36rs^3(s - 3r^2)$

Let's solve problem 12: $36k - 30$
# Explanation:
## Step 1: Find GCF of 36 and 30
GCF of 36 and 30 is 6.
## Step 2: Factor out 6
$36k\div6 = 6k$, $-30\div6=-5$
So $36k - 30 = 6(6k - 5)$
# Answer:
$6(6k - 5)$

Let's solve problem 13: $a^7b - a^{10}$
# Explanation:
## Step 1: Find GCF of terms
GCF of $a^7b$ and $a^{10}$ is $a^7$.
## Step 2: Factor out $a^7$
$a^7b\div a^7 = b$, $a^{10}\div a^7 = a^3$
So $a^7b - a^{10}=a^7(b - a^3)$
# Answer:
$a^7(b - a^3)$

Let's solve problem 14: $2c^3d^4 - 3c^4 + 4c^3$
# Explanation:
## Step 1: Find GCF of terms
GCF of $2c^3d^4$, $-3c^4$ and $4c^3$ is $c^3$.
## Step 2: Factor out $c^3$
$2c^3d^4\div c^3 = 2d^4$, $-3c^4\div c^3=-3c$, $4c^3\div c^3 = 4$
So $2c^3d^4 - 3c^4 + 4c^3 = c^3(2d^4 - 3c + 4)$
# Answer:
$c^3(2d^4 - 3c + 4)$

Let's solve problem 15: $3g^4 + 3g^7$
# Explanation:
## Step 1: Find GCF of terms
GCF of $3g^4$ and $3g^7$ is $3g^4$ (or $3g^7$, but factoring out $3g^4$ is better as the exponent is smaller).
## Step 2: Factor out $3g^4$
$3g^4\div3g^4 = 1$, $3g^7\div3g^4 = g^3$
So $3g^4 + 3g^7 = 3g^4(1 + g^3)$ (or $3g^7(\frac{1}{g^3}+1)$ but the first form is better)
# Answer:
$3g^4(1 + g^3)$ (or $3g^4(g^3 + 1)$)

Let's solve problem 16: $18x^5 - 48x^4 + 56x^3 - 86x$
# Explanation:
## Step 1: Find GCF of terms
GCF of 18, 48, 56, 86. Let's find GCF of coefficients:

18: 2×9

48: 2×24

56: 2×28

86: 2×43

So GCF of coefficients is 2.

GCF of variables: $x^5$, $x^4$, $x^3$, $x$ is $x$.

So overall GCF is $2x$.

## Step 2: Factor out $2x$
$18x^5\div2x = 9x^4$, $-48x^4\div2x=-24x^3$, $56x^3\div2x = 28x^2$, $-86x\div2x=-43$
So $18x^5 - 48x^4 + 56x^3 - 86x = 2x(9x^4 - 24x^3 + 28x^2 - 43)$
# Answer:
$2x(9x^4 - 24x^3 + 28x^2 - 43)$

Let's solve problem 17: $23y^{10} - 46y^7 + 68y^4 + 10y$
# Explanation:
## Step 1: Find GCF of terms
GCF of 23, 46, 68, 10. 23 is prime. GCF of 23, 46, 68, 10 is 1 (since 23 and 10 are coprime, 23 and 68 are coprime). Wait, 46 is 2×23, so GCF of 23, 46 is 23. But 23 and 68: 68÷23=2余22, so GCF of 23 and 68 is 1. So overall GCF of coefficients is 1.

GCF of variables: $y^{10}$, $y^7$, $y^4$, $y$ is $y$.

Wait, 23y¹⁰, -46y⁷, 68y⁴, 10y. Let's check coefficients again: 23, -46, 68, 10. GCF of 23 and 46 is 23, but 23 and 68: GCF(23,68)=1, 23 and 10: GCF(23,10)=1. So GCF of coefficients is 1. GCF of variables is $y$. So overall GCF is $y$? Wait, 23y¹⁰ ÷ y =23y⁹, -46y⁷ ÷ y=-46y⁶, 6

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QUESTION IMAGE

Question

Explanation:

Step 1: Find GCF of coefficients and variables

Step 2: Factor out GCF

Step 1: Find GCF of 12 and 16

Step 2: Factor out 4

Step 1: Find GCF of coefficients and variables

Step 2: Factor out \(-15d^3\)

Answer: