Factoring Difference of Two Squares - Free Printable
Educational worksheet: Factoring Difference of Two Squares. Download and print for classroom or home learning activities.
JPG
1811×2560
271.2 KB
Free · Personal Use
Quality Assured by Worksheets Library Team
Reviewed for educational accuracy and age-appropriateness
ID: #1391753
⭐
Show Answer Key & Explanations
Step-by-step solution for: Factoring Difference of Two Squares
▼
Show Answer Key & Explanations
Step-by-step solution for: Factoring Difference of Two Squares
Let's solve the factoring using GCF (Greatest Common Factor) worksheet step by step. We'll go through each expression in Section A and Section B, factoring them fully by identifying the greatest common factor (GCF) of all terms.
---
We look for the GCF of coefficients and variables, then factor it out.
---
1) 8x + 24
- GCF of 8 and 24 is 8
- No variable in both terms → only factor out 8
✔ $ 8(x + 3) $
---
2) 15 + 25y
- GCF of 15 and 25 is 5
- Variables: only 'y' in second term → no common variable
✔ $ 5(3 + 5y) $
---
3) 32 – 40w
- GCF of 32 and 40 is 8
- Only 'w' in second term → no common variable
✔ $ 8(4 - 5w) $
---
4) 18c – 36
- GCF of 18 and 36 is 18
- Only 'c' in first term → factor out 18
✔ $ 18(c - 2) $
---
5) 16d² – 4d
- GCF of 16 and 4 is 4
- Both have 'd', lowest power is $ d^1 $
✔ $ 4d(4d - 1) $
---
6) 12s + 60s²
- GCF of 12 and 60 is 12
- Both have 's', lowest power is $ s^1 $
✔ $ 12s(1 + 5s) $
---
7) 21xy + 14x
- GCF of 21 and 14 is 7
- Both have 'x' → factor x
- y is not in both → so GCF is $ 7x $
✔ $ 7x(3y + 2) $
---
8) 27ab – 18a²
- GCF of 27 and 18 is 9
- Both have 'a' → lowest power is $ a^1 $
- b is not in both → GCF is $ 9a $
✔ $ 9a(3b - 2a) $
---
9) 12s²t + 28s
- GCF of 12 and 28 is 4
- Both have 's' → lowest power $ s^1 $
- t only in first → GCF is $ 4s $
✔ $ 4s(3st + 7) $
---
10) 72wz + 45w²z
- GCF of 72 and 45 is 9
- Both have 'w' and 'z' → lowest powers: $ w^1, z^1 $
→ GCF = $ 9wz $
✔ $ 9wz(8 + 5w) $
---
11) 22x²y – 55xy²
- GCF of 22 and 55 is 11
- Both have $ x $ and $ y $ → lowest powers: $ x^1, y^1 $
→ GCF = $ 11xy $
✔ $ 11xy(2x - 5y) $
---
12) 16k³ + 24k²
- GCF of 16 and 24 is 8
- Both have $ k^2 $ → GCF = $ 8k^2 $
✔ $ 8k^2(2k + 3) $
---
13) 9h²g – 15h³
- GCF of 9 and 15 is 3
- Both have $ h^2 $ → GCF = $ 3h^2 $
✔ $ 3h^2(3g - 5h) $
---
14) 12c²d² + 20c³
- GCF of 12 and 20 is 4
- Both have $ c^2 $ → GCF = $ 4c^2 $
- d only in first → so GCF = $ 4c^2 $
✔ $ 4c^2(3d^2 + 5c) $
---
15) 28a³b² – 7a²b
- GCF of 28 and 7 is 7
- Both have $ a^2 $, $ b^1 $ → GCF = $ 7a^2b $
✔ $ 7a^2b(4ab - 1) $
---
16) 60x²y³ – 35xy²
- GCF of 60 and 35 is 5
- Both have $ x^1 $, $ y^2 $ → GCF = $ 5xy^2 $
✔ $ 5xy^2(12xy - 7) $
---
17) 88s⁴t + 56s³t²
- GCF of 88 and 56 is 8
- Both have $ s^3 $, $ t^1 $ → GCF = $ 8s^3t $
✔ $ 8s^3t(11s + 7t) $
---
18) 36p³q⁴ – 48p⁴q²
- GCF of 36 and 48 is 12
- Both have $ p^3 $, $ q^2 $ → GCF = $ 12p^3q^2 $
✔ $ 12p^3q^2(3q^2 - 4p) $
---
Now we may need to factor by grouping or find GCF across all terms.
---
1) 6 – 12gh + 3h
Rearrange: $ 6 + 3h - 12gh $
Group: $ (6 + 3h) - 12gh $
But better: Look for GCF of all terms?
- Coefficients: 6, 12, 3 → GCF = 3
- Variables: only 'h' appears in some, but not all → GCF = 3
Factor:
✔ $ 3(2 - 4gh + h) $
Note: Can't factor further unless rearranged.
Alternatively: $ 3(2 + h - 4gh) $
✔️ Final: $ \boxed{3(2 + h - 4gh)} $
---
2) 21st – 7t + 14
- GCF of 21, 7, 14 = 7
- No common variable in all → GCF = 7
✔ $ 7(3st - t + 2) $
---
3) 22 – 44vw + 11v
- GCF of 22, 44, 11 = 11
- Rearranged: $ 22 + 11v - 44vw $
→ All divisible by 11
✔ $ 11(2 + v - 4vw) $
---
4) 4ab + 2b – abc
Look at terms:
- 4ab, 2b, –abc
Common factors?
- All have 'b' → factor out b
→ $ b(4a + 2 - ac) $
Can’t factor more → ✔ $ b(4a + 2 - ac) $
---
5) 5suv – 10sv + 15su
All terms have 5s
- 5suv → 5s·uv
- –10sv → 5s·(-2v)
- 15su → 5s·3u
So GCF = 5s
✔ $ 5s(uv - 2v + 3u) $
---
6) 16xy + 24y – 8xyz
All terms divisible by 8y
- 16xy ÷ 8y = 2x
- 24y ÷ 8y = 3
- –8xyz ÷ 8y = –xz
✔ $ 8y(2x + 3 - xz) $
---
7) 9wu – 27wuv + 45w
All terms have 9w
- 9wu → 9w·u
- –27wuv → 9w·(-3uv)
- 45w → 9w·5
✔ $ 9w(u - 3uv + 5) $
---
8) 24gh – 12g + 15h
Check GCF: 24, 12, 15 → GCF = 3
No common variable in all → GCF = 3
✔ $ 3(8gh - 4g + 5h) $
---
9) 132pqr – 96qr + 108pqrs
Look at coefficients:
- 132, 96, 108 → GCF?
- 132 = 12×11, 96 = 12×8, 108 = 12×9 → GCF = 12
Variables:
- All have q and r?
- 132pqr → has p,q,r
- –96qr → has q,r
- 108pqrs → has p,q,r,s
→ So common variables: q, r
→ GCF = 12qr
Now divide:
- 132pqr ÷ 12qr = 11p
- –96qr ÷ 12qr = –8
- 108pqrs ÷ 12qr = 9ps
✔ $ 12qr(11p - 8 + 9ps) $
---
10) 2x + xy – x²
Rearranged: $ 2x + xy - x^2 $
All terms have x → factor out x
→ $ x(2 + y - x) $
✔ $ x(2 + y - x) $
---
11) 5k² – 10jk + k
All terms have k → factor out k
→ $ k(5k - 10j + 1) $
✔ $ k(5k - 10j + 1) $
---
12) 9cd – 3c²d + 12c
All terms have 3c
- 9cd ÷ 3c = 3d
- –3c²d ÷ 3c = –cd
- 12c ÷ 3c = 4
✔ $ 3c(3d - cd + 4) $
---
13) 7xyz + xy² – x²y
All terms have xy
- 7xyz → xy·7z
- xy² → xy·y
- –x²y → xy·(-x)
So GCF = xy
✔ $ xy(7z + y - x) $
---
14) e²f – 5e³f² + e²
All terms have e²
- e²f → e²·f
- –5e³f² → e²·(-5ef²)
- e² → e²·1
So GCF = e²
✔ $ e²(f - 5ef² + 1) $
---
15) 8st²u – 32s²t + 64st
Look at coefficients: 8, 32, 64 → GCF = 8
Variables:
- st²u, s²t, st → common: s and t → $ st $
→ GCF = 8st
Now divide:
- 8st²u ÷ 8st = tu
- –32s²t ÷ 8st = –4s
- 64st ÷ 8st = 8
✔ $ 8st(tu - 4s + 8) $
---
16) 12g³h – 9g²h² + 18g²h
Coefficients: 12, 9, 18 → GCF = 3
Variables:
- g³h, g²h², g²h → common: $ g^2 $, $ h $ → $ g^2h $
→ GCF = 3g²h
Divide:
- 12g³h ÷ 3g²h = 4g
- –9g²h² ÷ 3g²h = –3h
- 18g²h ÷ 3g²h = 6
✔ $ 3g^2h(4g - 3h + 6) $
---
17) ½ab + ¾a² – a
All terms have a → factor out a
But fractions → factor out a and simplify:
→ $ a\left(\frac{1}{2}b + \frac{3}{4}a - 1\right) $
To make nicer, factor out ¼a? Let's check:
But best: just factor a
✔ $ a\left(\frac{1}{2}b + \frac{3}{4}a - 1\right) $
Alternatively, multiply entire expression by 4 to eliminate fractions (but not required).
Since question asks to factor fully, this is acceptable.
---
18) ¾x⁴y – x²y³ + ½x³y²
All terms have x²y
Let’s check:
- ¾x⁴y → x²y·(¾x²)
- –x²y³ → x²y·(-y²)
- ½x³y² → x²y·(½x y)
So GCF = x²y
Now factor:
→ $ x^2y\left(\frac{3}{4}x^2 - y^2 + \frac{1}{2}xy\right) $
We can write it as:
✔ $ x^2y\left(\frac{3}{4}x^2 + \frac{1}{2}xy - y^2\right) $
Optional: Multiply inside by 4 to eliminate fractions:
But since it's factored, this is fine.
---
#### Section A
1. $ 8(x + 3) $
2. $ 5(3 + 5y) $
3. $ 8(4 - 5w) $
4. $ 18(c - 2) $
5. $ 4d(4d - 1) $
6. $ 12s(1 + 5s) $
7. $ 7x(3y + 2) $
8. $ 9a(3b - 2a) $
9. $ 4s(3st + 7) $
10. $ 9wz(8 + 5w) $
11. $ 11xy(2x - 5y) $
12. $ 8k^2(2k + 3) $
13. $ 3h^2(3g - 5h) $
14. $ 4c^2(3d^2 + 5c) $
15. $ 7a^2b(4ab - 1) $
16. $ 5xy^2(12xy - 7) $
17. $ 8s^3t(11s + 7t) $
18. $ 12p^3q^2(3q^2 - 4p) $
#### Section B
1. $ 3(2 + h - 4gh) $
2. $ 7(3st - t + 2) $
3. $ 11(2 + v - 4vw) $
4. $ b(4a + 2 - ac) $
5. $ 5s(uv - 2v + 3u) $
6. $ 8y(2x + 3 - xz) $
7. $ 9w(u - 3uv + 5) $
8. $ 3(8gh - 4g + 5h) $
9. $ 12qr(11p - 8 + 9ps) $
10. $ x(2 + y - x) $
11. $ k(5k - 10j + 1) $
12. $ 3c(3d - cd + 4) $
13. $ xy(7z + y - x) $
14. $ e^2(f - 5ef^2 + 1) $
15. $ 8st(tu - 4s + 8) $
16. $ 3g^2h(4g - 3h + 6) $
17. $ a\left(\frac{1}{2}b + \frac{3}{4}a - 1\right) $
18. $ x^2y\left(\frac{3}{4}x^2 + \frac{1}{2}xy - y^2\right) $
---
Let me know if you'd like these answers formatted into a printable sheet!
---
🔷 Section A: Factoring with Two Terms
We look for the GCF of coefficients and variables, then factor it out.
---
1) 8x + 24
- GCF of 8 and 24 is 8
- No variable in both terms → only factor out 8
✔ $ 8(x + 3) $
---
2) 15 + 25y
- GCF of 15 and 25 is 5
- Variables: only 'y' in second term → no common variable
✔ $ 5(3 + 5y) $
---
3) 32 – 40w
- GCF of 32 and 40 is 8
- Only 'w' in second term → no common variable
✔ $ 8(4 - 5w) $
---
4) 18c – 36
- GCF of 18 and 36 is 18
- Only 'c' in first term → factor out 18
✔ $ 18(c - 2) $
---
5) 16d² – 4d
- GCF of 16 and 4 is 4
- Both have 'd', lowest power is $ d^1 $
✔ $ 4d(4d - 1) $
---
6) 12s + 60s²
- GCF of 12 and 60 is 12
- Both have 's', lowest power is $ s^1 $
✔ $ 12s(1 + 5s) $
---
7) 21xy + 14x
- GCF of 21 and 14 is 7
- Both have 'x' → factor x
- y is not in both → so GCF is $ 7x $
✔ $ 7x(3y + 2) $
---
8) 27ab – 18a²
- GCF of 27 and 18 is 9
- Both have 'a' → lowest power is $ a^1 $
- b is not in both → GCF is $ 9a $
✔ $ 9a(3b - 2a) $
---
9) 12s²t + 28s
- GCF of 12 and 28 is 4
- Both have 's' → lowest power $ s^1 $
- t only in first → GCF is $ 4s $
✔ $ 4s(3st + 7) $
---
10) 72wz + 45w²z
- GCF of 72 and 45 is 9
- Both have 'w' and 'z' → lowest powers: $ w^1, z^1 $
→ GCF = $ 9wz $
✔ $ 9wz(8 + 5w) $
---
11) 22x²y – 55xy²
- GCF of 22 and 55 is 11
- Both have $ x $ and $ y $ → lowest powers: $ x^1, y^1 $
→ GCF = $ 11xy $
✔ $ 11xy(2x - 5y) $
---
12) 16k³ + 24k²
- GCF of 16 and 24 is 8
- Both have $ k^2 $ → GCF = $ 8k^2 $
✔ $ 8k^2(2k + 3) $
---
13) 9h²g – 15h³
- GCF of 9 and 15 is 3
- Both have $ h^2 $ → GCF = $ 3h^2 $
✔ $ 3h^2(3g - 5h) $
---
14) 12c²d² + 20c³
- GCF of 12 and 20 is 4
- Both have $ c^2 $ → GCF = $ 4c^2 $
- d only in first → so GCF = $ 4c^2 $
✔ $ 4c^2(3d^2 + 5c) $
---
15) 28a³b² – 7a²b
- GCF of 28 and 7 is 7
- Both have $ a^2 $, $ b^1 $ → GCF = $ 7a^2b $
✔ $ 7a^2b(4ab - 1) $
---
16) 60x²y³ – 35xy²
- GCF of 60 and 35 is 5
- Both have $ x^1 $, $ y^2 $ → GCF = $ 5xy^2 $
✔ $ 5xy^2(12xy - 7) $
---
17) 88s⁴t + 56s³t²
- GCF of 88 and 56 is 8
- Both have $ s^3 $, $ t^1 $ → GCF = $ 8s^3t $
✔ $ 8s^3t(11s + 7t) $
---
18) 36p³q⁴ – 48p⁴q²
- GCF of 36 and 48 is 12
- Both have $ p^3 $, $ q^2 $ → GCF = $ 12p^3q^2 $
✔ $ 12p^3q^2(3q^2 - 4p) $
---
🔷 Section B: Factoring with Three or More Terms
Now we may need to factor by grouping or find GCF across all terms.
---
1) 6 – 12gh + 3h
Rearrange: $ 6 + 3h - 12gh $
Group: $ (6 + 3h) - 12gh $
But better: Look for GCF of all terms?
- Coefficients: 6, 12, 3 → GCF = 3
- Variables: only 'h' appears in some, but not all → GCF = 3
Factor:
✔ $ 3(2 - 4gh + h) $
Note: Can't factor further unless rearranged.
Alternatively: $ 3(2 + h - 4gh) $
✔️ Final: $ \boxed{3(2 + h - 4gh)} $
---
2) 21st – 7t + 14
- GCF of 21, 7, 14 = 7
- No common variable in all → GCF = 7
✔ $ 7(3st - t + 2) $
---
3) 22 – 44vw + 11v
- GCF of 22, 44, 11 = 11
- Rearranged: $ 22 + 11v - 44vw $
→ All divisible by 11
✔ $ 11(2 + v - 4vw) $
---
4) 4ab + 2b – abc
Look at terms:
- 4ab, 2b, –abc
Common factors?
- All have 'b' → factor out b
→ $ b(4a + 2 - ac) $
Can’t factor more → ✔ $ b(4a + 2 - ac) $
---
5) 5suv – 10sv + 15su
All terms have 5s
- 5suv → 5s·uv
- –10sv → 5s·(-2v)
- 15su → 5s·3u
So GCF = 5s
✔ $ 5s(uv - 2v + 3u) $
---
6) 16xy + 24y – 8xyz
All terms divisible by 8y
- 16xy ÷ 8y = 2x
- 24y ÷ 8y = 3
- –8xyz ÷ 8y = –xz
✔ $ 8y(2x + 3 - xz) $
---
7) 9wu – 27wuv + 45w
All terms have 9w
- 9wu → 9w·u
- –27wuv → 9w·(-3uv)
- 45w → 9w·5
✔ $ 9w(u - 3uv + 5) $
---
8) 24gh – 12g + 15h
Check GCF: 24, 12, 15 → GCF = 3
No common variable in all → GCF = 3
✔ $ 3(8gh - 4g + 5h) $
---
9) 132pqr – 96qr + 108pqrs
Look at coefficients:
- 132, 96, 108 → GCF?
- 132 = 12×11, 96 = 12×8, 108 = 12×9 → GCF = 12
Variables:
- All have q and r?
- 132pqr → has p,q,r
- –96qr → has q,r
- 108pqrs → has p,q,r,s
→ So common variables: q, r
→ GCF = 12qr
Now divide:
- 132pqr ÷ 12qr = 11p
- –96qr ÷ 12qr = –8
- 108pqrs ÷ 12qr = 9ps
✔ $ 12qr(11p - 8 + 9ps) $
---
10) 2x + xy – x²
Rearranged: $ 2x + xy - x^2 $
All terms have x → factor out x
→ $ x(2 + y - x) $
✔ $ x(2 + y - x) $
---
11) 5k² – 10jk + k
All terms have k → factor out k
→ $ k(5k - 10j + 1) $
✔ $ k(5k - 10j + 1) $
---
12) 9cd – 3c²d + 12c
All terms have 3c
- 9cd ÷ 3c = 3d
- –3c²d ÷ 3c = –cd
- 12c ÷ 3c = 4
✔ $ 3c(3d - cd + 4) $
---
13) 7xyz + xy² – x²y
All terms have xy
- 7xyz → xy·7z
- xy² → xy·y
- –x²y → xy·(-x)
So GCF = xy
✔ $ xy(7z + y - x) $
---
14) e²f – 5e³f² + e²
All terms have e²
- e²f → e²·f
- –5e³f² → e²·(-5ef²)
- e² → e²·1
So GCF = e²
✔ $ e²(f - 5ef² + 1) $
---
15) 8st²u – 32s²t + 64st
Look at coefficients: 8, 32, 64 → GCF = 8
Variables:
- st²u, s²t, st → common: s and t → $ st $
→ GCF = 8st
Now divide:
- 8st²u ÷ 8st = tu
- –32s²t ÷ 8st = –4s
- 64st ÷ 8st = 8
✔ $ 8st(tu - 4s + 8) $
---
16) 12g³h – 9g²h² + 18g²h
Coefficients: 12, 9, 18 → GCF = 3
Variables:
- g³h, g²h², g²h → common: $ g^2 $, $ h $ → $ g^2h $
→ GCF = 3g²h
Divide:
- 12g³h ÷ 3g²h = 4g
- –9g²h² ÷ 3g²h = –3h
- 18g²h ÷ 3g²h = 6
✔ $ 3g^2h(4g - 3h + 6) $
---
17) ½ab + ¾a² – a
All terms have a → factor out a
But fractions → factor out a and simplify:
→ $ a\left(\frac{1}{2}b + \frac{3}{4}a - 1\right) $
To make nicer, factor out ¼a? Let's check:
But best: just factor a
✔ $ a\left(\frac{1}{2}b + \frac{3}{4}a - 1\right) $
Alternatively, multiply entire expression by 4 to eliminate fractions (but not required).
Since question asks to factor fully, this is acceptable.
---
18) ¾x⁴y – x²y³ + ½x³y²
All terms have x²y
Let’s check:
- ¾x⁴y → x²y·(¾x²)
- –x²y³ → x²y·(-y²)
- ½x³y² → x²y·(½x y)
So GCF = x²y
Now factor:
→ $ x^2y\left(\frac{3}{4}x^2 - y^2 + \frac{1}{2}xy\right) $
We can write it as:
✔ $ x^2y\left(\frac{3}{4}x^2 + \frac{1}{2}xy - y^2\right) $
Optional: Multiply inside by 4 to eliminate fractions:
But since it's factored, this is fine.
---
✔ Final Answers Summary
#### Section A
1. $ 8(x + 3) $
2. $ 5(3 + 5y) $
3. $ 8(4 - 5w) $
4. $ 18(c - 2) $
5. $ 4d(4d - 1) $
6. $ 12s(1 + 5s) $
7. $ 7x(3y + 2) $
8. $ 9a(3b - 2a) $
9. $ 4s(3st + 7) $
10. $ 9wz(8 + 5w) $
11. $ 11xy(2x - 5y) $
12. $ 8k^2(2k + 3) $
13. $ 3h^2(3g - 5h) $
14. $ 4c^2(3d^2 + 5c) $
15. $ 7a^2b(4ab - 1) $
16. $ 5xy^2(12xy - 7) $
17. $ 8s^3t(11s + 7t) $
18. $ 12p^3q^2(3q^2 - 4p) $
#### Section B
1. $ 3(2 + h - 4gh) $
2. $ 7(3st - t + 2) $
3. $ 11(2 + v - 4vw) $
4. $ b(4a + 2 - ac) $
5. $ 5s(uv - 2v + 3u) $
6. $ 8y(2x + 3 - xz) $
7. $ 9w(u - 3uv + 5) $
8. $ 3(8gh - 4g + 5h) $
9. $ 12qr(11p - 8 + 9ps) $
10. $ x(2 + y - x) $
11. $ k(5k - 10j + 1) $
12. $ 3c(3d - cd + 4) $
13. $ xy(7z + y - x) $
14. $ e^2(f - 5ef^2 + 1) $
15. $ 8st(tu - 4s + 8) $
16. $ 3g^2h(4g - 3h + 6) $
17. $ a\left(\frac{1}{2}b + \frac{3}{4}a - 1\right) $
18. $ x^2y\left(\frac{3}{4}x^2 + \frac{1}{2}xy - y^2\right) $
---
Let me know if you'd like these answers formatted into a printable sheet!
Parent Tip: Review the logic above to help your child master the concept of factoring difference of two squares worksheet.
