Algebra 2 worksheet for factoring polynomials by finding the greatest common factor.

Here are the solutions for each problem on the worksheet. I have factored out the Greatest Common Factor (GCF) for each expression. If an expression cannot be factored further using integers, it is marked as PRIME.

1) $5ax - 5a$
- The GCF of $5ax$ and $5a$ is $5a$.
- Divide each term by $5a$: $\frac{5ax}{5a} = x$ and $\frac{-5a}{5a} = -1$.
- Answer: $5a(x - 1)$

2) $5xz + 2xy - 3yz$
- Look at the variables: $x$ is in the first two terms but not the third. $y$ is in the last two but not the first. $z$ is in the first and last but not the second.
- Look at the numbers: 5, 2, and 3 share no common factor other than 1.
- There is no common factor for all three terms.
- Answer: PRIME

3) $24ab^6 + 12ab^3 - 18ab^7$
- Numbers: GCF of 24, 12, 18 is 6.
- Variables: All terms have at least one $a$ and at least $b^3$. So the variable part is $ab^3$.
- Total GCF: $6ab^3$.
- Divide terms: $\frac{24ab^6}{6ab^3} = 4b^3$, $\frac{12ab^3}{6ab^3} = 2$, $\frac{-18ab^7}{6ab^3} = -3b^4$.
- Answer: $6ab^3(4b^3 + 2 - 3b^4)$

4) $3n^2 + 9$
- The GCF of 3 and 9 is 3.
- Divide terms: $\frac{3n^2}{3} = n^2$, $\frac{9}{3} = 3$.
- Answer: $3(n^2 + 3)$

5) $x(x + y) - y(x + y)$
- Both parts contain the group $(x + y)$. This is the common factor.
- Pull out $(x + y)$: What is left is $x$ from the first part and $-y$ from the second part.
- Result: $(x + y)(x - y)$.
- Answer: $(x + y)(x - y)$

6) $25k^3 + 20k^2 + 10k$
- Numbers: GCF of 25, 20, 10 is 5.
- Variables: Lowest power of $k$ is $k^1$ (or just $k$).
- Total GCF: $5k$.
- Divide terms: $\frac{25k^3}{5k} = 5k^2$, $\frac{20k^2}{5k} = 4k$, $\frac{10k}{5k} = 2$.
- Answer: $5k(5k^2 + 4k + 2)$

7) $8x^2 + 5x - 7$
- Numbers: 8, 5, 7 share no common factor.
- Variables: The last term (-7) has no $x$, so we cannot factor out an $x$.
- Answer: PRIME

8) $7ab^2 - 58ab$
- Numbers: 7 and 58 share no common factor (7 is prime, 58 is $2 \times 29$).
- Variables: Both have $a$ and at least one $b$. GCF is $ab$.
- Divide terms: $\frac{7ab^2}{ab} = 7b$, $\frac{-58ab}{ab} = -58$.
- Answer: $ab(7b - 58)$

9) $mnx^2 - nx^2 + m^2x$
- Variables: First two terms have $x^2$, last has $x$. So we can pull out $x$.
- Check coefficients/other vars: The middle term $-nx^2$ does not have an $m$. The last term $m^2x$ does not have an $n$. So we can only pull out $x$.
- GCF: $x$.
- Divide terms: $mnx - nx + m^2$.
- Answer: $x(mnx - nx + m^2)$

10) $x^3(x^2 - 5) + 8(x^2 - 5)$
- Both parts contain the group $(x^2 - 5)$.
- Pull out $(x^2 - 5)$. What is left is $x^3$ and $+8$.
- Answer: $(x^2 - 5)(x^3 + 8)$

11) $6k^3 - 18k^4$
- Numbers: GCF of 6 and 18 is 6.
- Variables: Lowest power is $k^3$.
- Total GCF: $6k^3$.
- Divide terms: $\frac{6k^3}{6k^3} = 1$, $\frac{-18k^4}{6k^3} = -3k$.
- Answer: $6k^3(1 - 3k)$

12) $12m^3 - 8m^2 + 20m^5$
- Numbers: GCF of 12, 8, 20 is 4.
- Variables: Lowest power is $m^2$.
- Total GCF: $4m^2$.
- Divide terms: $\frac{12m^3}{4m^2} = 3m$, $\frac{-8m^2}{4m^2} = -2$, $\frac{20m^5}{4m^2} = 5m^3$.
- Answer: $4m^2(3m - 2 + 5m^3)$

13) $8xy - 6xz - 8x$
- Numbers: GCF of 8, 6, 8 is 2.
- Variables: All terms have an $x$.
- Total GCF: $2x$.
- Divide terms: $\frac{8xy}{2x} = 4y$, $\frac{-6xz}{2x} = -3z$, $\frac{-8x}{2x} = -4$.
- Answer: $2x(4y - 3z - 4)$

14) $3x^4 + 12x^2 - 33$
- Numbers: GCF of 3, 12, 33 is 3.
- Variables: The last term (-33) has no $x$, so no variable can be factored out.
- Total GCF: 3.
- Divide terms: $x^4 + 4x^2 - 11$.
- Answer: $3(x^4 + 4x^2 - 11)$

15) $8a^5b^2 - 28a^3b^3 + 4a^3b^2$
- Numbers: GCF of 8, 28, 4 is 4.
- Variables for $a$: Lowest power is $a^3$.
- Variables for $b$: Lowest power is $b^2$.
- Total GCF: $4a^3b^2$.
- Divide terms: $\frac{8a^5b^2}{4a^3b^2} = 2a^2$, $\frac{-28a^3b^3}{4a^3b^2} = -7b$, $\frac{4a^3b^2}{4a^3b^2} = 1$.
- Answer: $4a^3b^2(2a^2 - 7b + 1)$

16) $4k^3 + 18k^2 - 6k^4$
- Numbers: GCF of 4, 18, 6 is 2.
- Variables: Lowest power is $k^2$.
- Total GCF: $2k^2$.
- Divide terms: $\frac{4k^3}{2k^2} = 2k$, $\frac{18k^2}{2k^2} = 9$, $\frac{-6k^4}{2k^2} = -3k^2$.
- Answer: $2k^2(2k + 9 - 3k^2)$

Final Answer:
1) $5a(x - 1)$
2) PRIME
3) $6ab^3(4b^3 + 2 - 3b^4)$
4) $3(n^2 + 3)$
5) $(x + y)(x - y)$
6) $5k(5k^2 + 4k + 2)$
7) PRIME
8) $ab(7b - 58)$
9) $x(mnx - nx + m^2)$
10) $(x^2 - 5)(x^3 + 8)$
11) $6k^3(1 - 3k)$
12) $4m^2(3m - 2 + 5m^3)$
13) $2x(4y - 3z - 4)$
14) $3(x^4 + 4x^2 - 11)$
15) $4a^3b^2(2a^2 - 7b + 1)$
16) $2k^2(2k + 9 - 3k^2)$