Here are the correct solutions for the factoring problems on your worksheet. I have worked through each one step-by-step to ensure accuracy.
Set 1: Greatest Common Factor (GCF)
*Goal: Find the largest number and variable that divides evenly into every term.*
1. $21x - 15y$
* Numbers: The GCF of 21 and 15 is
3.
* Variables: There are no shared variables ($x$ and $y$ are different).
* Factor out 3: $3(7x - 5y)$
2. $14c^3 + 2c$
* Numbers: The GCF of 14 and 2 is
2.
* Variables: Both terms have $c$. The lowest power is $c^1$ (just $c$). So, factor out
$2c$.
* Divide terms by $2c$: $\frac{14c^3}{2c} = 7c^2$ and $\frac{2c}{2c} = 1$.
* Answer: $2c(7c^2 + 1)$
3. $5p^3 + 12q^2$
* Numbers: 5 and 12 share no common factors other than 1.
* Variables: $p$ and $q$ are different.
* Answer:
Cannot be factored (Prime)
4. $8x^3y^4 - 22x^2y^6$
* Numbers: The GCF of 8 and 22 is
2.
* Variables:
* $x$: Lowest power is $x^2$.
* $y$: Lowest power is $y^4$.
* GCF is
$2x^2y^4$.
* Divide terms:
* $\frac{8x^3y^4}{2x^2y^4} = 4x$
* $\frac{-22x^2y^6}{2x^2y^4} = -11y^2$
* Answer: $2x^2y^4(4x - 11y^2)$
5. $10a^2b^3 + 9ab^2 - a^2b$
* Numbers: 10, 9, and 1 share no common factor.
* Variables:
* $a$: Lowest power is $a^1$ ($a$).
* $b$: Lowest power is $b^1$ ($b$).
* GCF is
$ab$.
* Divide terms:
* $\frac{10a^2b^3}{ab} = 10ab^2$
* $\frac{9ab^2}{ab} = 9b$
* $\frac{-a^2b}{ab} = -a$
* Answer: $ab(10ab^2 + 9b - a)$
6. $21m^3n^2 + 6m^2n + 15m^2$
* Numbers: The GCF of 21, 6, and 15 is
3.
* Variables:
* $m$: All terms have at least $m^2$.
* $n$: The last term ($15m^2$) has no $n$, so we cannot factor out any $n$.
* GCF is
$3m^2$.
* Divide terms:
* $\frac{21m^3n^2}{3m^2} = 7mn^2$
* $\frac{6m^2n}{3m^2} = 2n$
* $\frac{15m^2}{3m^2} = 5$
* Answer: $3m^2(7mn^2 + 2n + 5)$
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Set 2: Difference of Squares
*Formula: $a^2 - b^2 = (a - b)(a + b)$*
*Note: You can only use this if there is a minus sign between two perfect squares.*
7. $a^2 - 64$
* Square roots: $\sqrt{a^2} = a$ and $\sqrt{64} = 8$.
* Answer: $(a - 8)(a + 8)$
8. $y^2 - 289$
* Square roots: $\sqrt{y^2} = y$ and $\sqrt{289} = 17$.
* Answer: $(y - 17)(y + 17)$
9. $m^2 + 81$
* This is a
sum of squares (plus sign), not a difference. It cannot be factored using real numbers.
* Answer:
Cannot be factored (Prime)
10. $1 - 25n^2$
* Square roots: $\sqrt{1} = 1$ and $\sqrt{25n^2} = 5n$.
* Answer: $(1 - 5n)(1 + 5n)$
11. $16x^2 - 49y^2$
* Square roots: $\sqrt{16x^2} = 4x$ and $\sqrt{49y^2} = 7y$.
* Answer: $(4x - 7y)(4x + 7y)$
12. $w^4 - 100$
* Square roots: $\sqrt{w^4} = w^2$ and $\sqrt{100} = 10$.
* Answer: $(w^2 - 10)(w^2 + 10)$
*(Note: $w^2 - 10$ cannot be factored further because 10 is not a perfect square).*
13. $c^2 - 81d^2$
* Square roots: $\sqrt{c^2} = c$ and $\sqrt{81d^2} = 9d$.
* Answer: $(c - 9d)(c + 9d)$
14. $196a^2 - b^2$
* Square roots: $\sqrt{196a^2} = 14a$ and $\sqrt{b^2} = b$.
* Answer: $(14a - b)(14a + b)$
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Final Answer:
Set 1:
1. $3(7x - 5y)$
2. $2c(7c^2 + 1)$
3. Cannot be factored
4. $2x^2y^4(4x - 11y^2)$
5. $ab(10ab^2 + 9b - a)$
6. $3m^2(7mn^2 + 2n + 5)$
Set 2:
7. $(a - 8)(a + 8)$
8. $(y - 17)(y + 17)$
9. Cannot be factored
10. $(1 - 5n)(1 + 5n)$
11. $(4x - 7y)(4x + 7y)$
12. $(w^2 - 10)(w^2 + 10)$
13. $(c - 9d)(c + 9d)$
14. $(14a - b)(14a + b)$
Parent Tip: Review the logic above to help your child master the concept of factoring review worksheet answers.
