Let's solve each problem on the
Factoring Binomials Worksheet step by step. We'll factor each expression completely.
---
1. $ 2x^2 + 28x $
- Both terms have a common factor of $ 2x $
- Factor out $ 2x $:
$$
2x(x + 14)
$$
✔ Answer: $ 2x(x + 14) $
---
2. $ 19x - 57x $
- Combine like terms first:
$ 19x - 57x = -38x $
- This is a monomial, not a binomial anymore.
- But if we interpret it as factoring the
original expression (even though it simplifies), we can factor out $ x $:
$$
x(19 - 57) = x(-38) = -38x
$$
- So, factored form is just $ -38x $
But since it's asking to factor *completely*, and there’s only one term after simplification, we write:
✔ Answer: $ -38x $
> Note: The expression simplifies to a single term, so factoring gives $ -38x $, which is fully factored.
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3. $ 27a^3b^3 + 36a^4b^2 $
- Find GCF of coefficients: $ \gcd(27, 36) = 9 $
- Variables:
- $ a^3 $ and $ a^4 $ → lowest power is $ a^3 $
- $ b^3 $ and $ b^2 $ → lowest power is $ b^2 $
- So GCF is $ 9a^3b^2 $
- Factor it out:
$$
9a^3b^2(3b + 4a)
$$
✔ Answer: $ 9a^3b^2(3b + 4a) $
---
4. $ -16x + 20x^3 $
- Rewrite in standard order: $ 20x^3 - 16x $
- GCF: $ 4x $ (since $ \gcd(20,16)=4 $, and both have $ x $)
- Factor out $ 4x $:
$$
4x(5x^2 - 4)
$$
✔ Answer: $ 4x(5x^2 - 4) $
---
5. $ 12y^3 - 18y^5 $
- GCF: $ \gcd(12,18) = 6 $, and $ y^3 $ is common
- So GCF is $ 6y^3 $
- Factor:
$$
6y^3(2 - 3y^2)
$$
✔ Answer: $ 6y^3(2 - 3y^2) $
---
6. $ x^2 - 25 $
- This is a
difference of squares:
$ a^2 - b^2 = (a - b)(a + b) $
- $ x^2 - 25 = x^2 - 5^2 $
- So:
$$
(x - 5)(x + 5)
$$
✔ Answer: $ (x - 5)(x + 5) $
---
7. $ 3x^4 + 9x^2 $
- GCF: $ 3x^2 $
- Factor:
$$
3x^2(x^2 + 3)
$$
✔ Answer: $ 3x^2(x^2 + 3) $
---
8. $ 12x^2y - 42xyz $
- GCF of coefficients: $ \gcd(12,42) = 6 $
- Variables:
- $ x^2 $ and $ x $ → $ x $
- $ y $ and $ y $ → $ y $
- $ z $ is only in second term → not common
- So GCF is $ 6xy $
- Factor:
$$
6xy(2x - 7z)
$$
✔ Answer: $ 6xy(2x - 7z) $
---
9. $ 9x^2 + 5x $
- GCF: $ x $
- Factor:
$$
x(9x + 5)
$$
✔ Answer: $ x(9x + 5) $
---
10. $ 3x^2 - 108 $
- First, factor out GCF: $ 3 $
$$
3(x^2 - 36)
$$
- Now, $ x^2 - 36 $ is a difference of squares: $ x^2 - 6^2 $
$$
3(x - 6)(x + 6)
$$
✔ Answer: $ 3(x - 6)(x + 6) $
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✔ Final Answers Summary:
| Problem | Answer |
|--------|--------|
| 1 | $ 2x(x + 14) $ |
| 2 | $ -38x $ |
| 3 | $ 9a^3b^2(3b + 4a) $ |
| 4 | $ 4x(5x^2 - 4) $ |
| 5 | $ 6y^3(2 - 3y^2) $ |
| 6 | $ (x - 5)(x + 5) $ |
| 7 | $ 3x^2(x^2 + 3) $ |
| 8 | $ 6xy(2x - 7z) $ |
| 9 | $ x(9x + 5) $ |
| 10 | $ 3(x - 6)(x + 6) $ |
Let me know if you'd like these explained further or formatted for printing!
Parent Tip: Review the logic above to help your child master the concept of binomial worksheet.
