Factoring out the GCF worksheet with ten algebraic expressions to simplify by finding the greatest common factor.

Here are the step-by-step solutions for factoring out the Greatest Common Factor (GCF) for each problem.

1. $15x^3 - 25x^2 + 55x$
* Coefficients: The numbers are 15, 25, and 55. They are all divisible by 5.
* Variables: Each term has at least one $x$. The lowest power is $x^1$ (or just $x$).
* GCF: $5x$
* Divide:
* $15x^3 \div 5x = 3x^2$
* $-25x^2 \div 5x = -5x$
* $55x \div 5x = 11$
* Result: $5x(3x^2 - 5x + 11)$

2. $36x^3 + 63x^2 - 27x$
* Coefficients: 36, 63, and 27 are all divisible by 9.
* Variables: Each term has an $x$. The lowest power is $x$.
* GCF: $9x$
* Divide:
* $36x^3 \div 9x = 4x^2$
* $63x^2 \div 9x = 7x$
* $-27x \div 9x = -3$
* Result: $9x(4x^2 + 7x - 3)$

3. $25x^{12} - 35x^6$
* Coefficients: 25 and 35 are both divisible by 5.
* Variables: We have $x^{12}$ and $x^6$. The lowest power is $x^6$.
* GCF: $5x^6$
* Divide:
* $25x^{12} \div 5x^6 = 5x^6$ (subtract exponents: $12-6=6$)
* $-35x^6 \div 5x^6 = -7$
* Result: $5x^6(5x^6 - 7)$

4. $-6p^5 - 6p^4$
* Coefficients: Both are divisible by 6. Since the first term is negative, we usually factor out a negative number to make the inside positive. So, we use $-6$.
* Variables: Lowest power of $p$ is $p^4$.
* GCF: $-6p^4$
* Divide:
* $-6p^5 \div -6p^4 = p$
* $-6p^4 \div -6p^4 = 1$
* Result: $-6p^4(p + 1)$

5. $72x^5 - 72x^3 - 80x^2$
* Coefficients: 72, 72, and 80. They are all even, so divide by 2 first: 36, 36, 40. Divide by 2 again: 18, 18, 20. Divide by 2 again: 9, 9, 10. No more common numbers. So, $2 \times 2 \times 2 = 8$.
* Variables: Lowest power is $x^2$.
* GCF: $8x^2$
* Divide:
* $72x^5 \div 8x^2 = 9x^3$
* $-72x^3 \div 8x^2 = -9x$
* $-80x^2 \div 8x^2 = -10$
* Result: $8x^2(9x^3 - 9x - 10)$

6. $6x^2yz + 2xy^2z - 4xyz$
* Coefficients: 6, 2, and 4 are all divisible by 2.
* Variables:
* $x$: powers are 2, 1, 1. Lowest is $x$.
* $y$: powers are 1, 2, 1. Lowest is $y$.
* $z$: powers are 1, 1, 1. Lowest is $z$.
* GCF: $2xyz$
* Divide:
* $6x^2yz \div 2xyz = 3x$
* $2xy^2z \div 2xyz = y$
* $-4xyz \div 2xyz = -2$
* Result: $2xyz(3x + y - 2)$

7. $-16p^3q^2 + 24p^2q^3 - 32p^4q$
* Coefficients: 16, 24, 32. All divisible by 8. First term is negative, so use $-8$.
* Variables:
* $p$: powers are 3, 2, 4. Lowest is $p^2$.
* $q$: powers are 2, 3, 1. Lowest is $q$.
* GCF: $-8p^2q$
* Divide:
* $-16p^3q^2 \div -8p^2q = 2pq$
* $24p^2q^3 \div -8p^2q = -3q^2$
* $-32p^4q \div -8p^2q = 4p^2$
* Result: $-8p^2q(2pq - 3q^2 + 4p^2)$

8. $7wx(a - 9) - 10w(9 - a)$
* Observation: The terms $(a - 9)$ and $(9 - a)$ are opposites. We can rewrite $(9 - a)$ as $-1(a - 9)$.
* Rewrite: $7wx(a - 9) - 10w[-1(a - 9)]$ becomes $7wx(a - 9) + 10w(a - 9)$.
* Common Factors: Both terms now have $w$ and $(a - 9)$.
* GCF: $w(a - 9)$
* Divide:
* First part leaves: $7x$
* Second part leaves: $+10$
* Result: $w(a - 9)(7x + 10)$

9. $36w^5 + 40w^4 - 160w^3 - 20w^2$
* Coefficients: 36, 40, 160, 20. All end in 0 or are even, so divisible by 2. Let's check 4.
* $36 \div 4 = 9$
* $40 \div 4 = 10$
* $160 \div 4 = 40$
* $20 \div 4 = 5$
* Can we go higher? 9 and 5 share no factors. So, 4 is the GCF for numbers.
* Variables: Lowest power is $w^2$.
* GCF: $4w^2$
* Divide:
* $36w^5 \div 4w^2 = 9w^3$
* $40w^4 \div 4w^2 = 10w^2$
* $-160w^3 \div 4w^2 = -40w$
* $-20w^2 \div 4w^2 = -5$
* Result: $4w^2(9w^3 + 10w^2 - 40w - 5)$

10. $bc(m + 10) - 5b(10 + m)$
* Observation: $(m + 10)$ is the same as $(10 + m)$.
* Common Factors: Both terms have $b$ and the group $(m + 10)$.
* GCF: $b(m + 10)$
* Divide:
* First part leaves: $c$
* Second part leaves: $-5$
* Result: $b(m + 10)(c - 5)$

Final Answer:
1. $5x(3x^2 - 5x + 11)$
2. $9x(4x^2 + 7x - 3)$
3. $5x^6(5x^6 - 7)$
4. $-6p^4(p + 1)$
5. $8x^2(9x^3 - 9x - 10)$
6. $2xyz(3x + y - 2)$
7. $-8p^2q(2pq - 3q^2 + 4p^2)$
8. $w(a - 9)(7x + 10)$
9. $4w^2(9w^3 + 10w^2 - 40w - 5)$
10. $b(m + 10)(c - 5)$