Find the G.C.F. for Monomials - Homework Questions
Homework questions worksheet for finding the G.C.F. of monomials with eight problems listed, each containing algebraic expressions with variables and exponents.
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Step-by-step solution for: Greatest common factor Homework questions Monomials
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Show Answer Key & Explanations
Step-by-step solution for: Greatest common factor Homework questions Monomials
You're asking to find the Greatest Common Factor (G.C.F.) — also called the Greatest Common Divisor (G.C.D.) — of sets of monomials.
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1. Factor each monomial into its prime factors (including variables with exponents).
2. For each variable, take the lowest exponent that appears in all monomials.
3. Multiply those together — that’s your G.C.F.
4. If a variable doesn’t appear in all monomials, it’s not included in the G.C.F.
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Let’s solve each problem step by step:
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## 🔢 LEFT COLUMN
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- Break down:
- `a²bc` → a² · b · c
- `b²c` → b² · c
- `c²a` → c² · a
- Variables present in all three: only `c` appears in all.
- Lowest exponent of `c`: min(1, 1, 2) = 1
- `a` is missing in `b²c` → not included
- `b` is missing in `c²a` → not included
✔ G.C.F. = c
> ❗ You wrote “_a” — that’s incorrect. The correct answer is c.
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- Break down:
- `p²q²s²` → p² · q² · s²
- `q²r²s²` → q² · r² · s²
- `pqr` → p · q · r
- Variables common to all? Let’s check:
- `p`: in first and third → but NOT in second → ✘
- `q`: in all → min exponent: min(2,2,1) = 1
- `r`: in second and third → NOT in first → ✘
- `s`: in first and second → NOT in third → ✘
✔ Only `q` is common → G.C.F. = q
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Assuming case-insensitive or typo — probably meant `qrt`, `rto`, `pqr` (all lowercase).
- `qrt` → q · r · t
- `rto` → r · t · o
- `pqr` → p · q · r
Common variables? Only `r` appears in all three.
✔ G.C.F. = r
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Rewrite for clarity: `wxy`, `x²y²`, `w x y³`
- All have `x` and `y`
- `x`: min(1,2,1) = 1
- `y`: min(1,2,3) = 1
- `w`: not in second → ✘
✔ G.C.F. = xy
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Write as: `m n o`, `m² n²`, `m n² o²`
- All have `m` and `n`
- `m`: min(1,2,1) = 1
- `n`: min(1,2,2) = 1
- `o`: not in second → ✘
✔ G.C.F. = mn
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Break down:
- `a²b²m²` → a² b² m²
- `m²n²` → m² n²
- `b²n²` → b² n²
Common variables? None appear in all three!
- `a` → only in first
- `b` → in first and third, not second
- `m` → in first and second, not third
- `n` → in second and third, not first
✔ No common variable → G.C.F. = 1
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All have `a`, `b`, `c`
- `a`: min(3,1,2) = 1
- `b`: min(3,1,2) = 1
- `c`: min(3,1,2) = 1
✔ G.C.F. = abc
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All have `x`, `y`, `z`
- `x`: min(3,1,2) = 1
- `y`: min(3,1,2) = 1
- `z`: min(3,1,2) = 1
✔ G.C.F. = xyz
---
## 🔢 RIGHT COLUMN
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→ `a b² c`, `b² c`, `a² b`
Common variables? Only `b` appears in all.
- `b`: min(2,2,1) = 1
✔ G.C.F. = b
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→ All have `m` and `n`
- `m`: min(2,2,1) = 1
- `n`: min(2,2,1) = 1
- `z`, `y` → not in all
✔ G.C.F. = mn
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→ `p r t`, `p r o`, `p q r`
Common variables? `p` and `r` are in all.
- `p`: min(1,1,1) = 1
- `r`: min(1,1,1) = 1
✔ G.C.F. = pr
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→ `a b c`, `b² c²`, `a³ b c`
All have `b` and `c`
- `b`: min(1,2,1) = 1
- `c`: min(1,2,1) = 1
- `a`: not in second → ✘
✔ G.C.F. = bc
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→ `x z o`, `x² z²`, `x² z² m`
Common? Only `x` and `z`
- `x`: min(1,2,2) = 1
- `z`: min(1,2,2) = 1
✔ G.C.F. = xz
---
All have `u²v²`
- `u`: min(2,2,2) = 2
- `v`: min(2,2,2) = 2
- `t`: only in first → ✘
✔ G.C.F. = u²v²
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All have `m`, `n`, `a`
- `m`: min(3,1,2) = 1
- `n`: min(3,1,2) = 1
- `a`: min(3,1,2) = 1
✔ G.C.F. = amn
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All have `a`, `b`, `c`
- `a`: min(3,1,2) = 1
- `b`: min(3,1,2) = 1
- `c`: min(3,1,2) = 1
✔ G.C.F. = abc
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## 📝 FINAL ANSWERS:
1. c
2. q
3. r
4. xy
5. mn
6. 1
7. abc
8. xyz
1. b
2. mn
3. pr
4. bc
5. xz
6. u²v²
7. amn
8. abc
---
✔ Note: In problem #1 left column, you wrote “_a” — that’s incorrect. The correct G.C.F. is c.
Let me know if you want this formatted as a worksheet or need explanations for any specific one!
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✔ How to Find G.C.F. of Monomials:
1. Factor each monomial into its prime factors (including variables with exponents).
2. For each variable, take the lowest exponent that appears in all monomials.
3. Multiply those together — that’s your G.C.F.
4. If a variable doesn’t appear in all monomials, it’s not included in the G.C.F.
---
Let’s solve each problem step by step:
---
## 🔢 LEFT COLUMN
---
1. `a²bc`, `b²c`, `c²a`
- Break down:
- `a²bc` → a² · b · c
- `b²c` → b² · c
- `c²a` → c² · a
- Variables present in all three: only `c` appears in all.
- Lowest exponent of `c`: min(1, 1, 2) = 1
- `a` is missing in `b²c` → not included
- `b` is missing in `c²a` → not included
✔ G.C.F. = c
> ❗ You wrote “_a” — that’s incorrect. The correct answer is c.
---
2. `p²q²s²`, `q²r²s²`, `pqr`
- Break down:
- `p²q²s²` → p² · q² · s²
- `q²r²s²` → q² · r² · s²
- `pqr` → p · q · r
- Variables common to all? Let’s check:
- `p`: in first and third → but NOT in second → ✘
- `q`: in all → min exponent: min(2,2,1) = 1
- `r`: in second and third → NOT in first → ✘
- `s`: in first and second → NOT in third → ✘
✔ Only `q` is common → G.C.F. = q
---
3. `Qrt`, `rto`, `pqr`
Assuming case-insensitive or typo — probably meant `qrt`, `rto`, `pqr` (all lowercase).
- `qrt` → q · r · t
- `rto` → r · t · o
- `pqr` → p · q · r
Common variables? Only `r` appears in all three.
✔ G.C.F. = r
---
4. `wxy`, `x²y²`, `xwy³`
Rewrite for clarity: `wxy`, `x²y²`, `w x y³`
- All have `x` and `y`
- `x`: min(1,2,1) = 1
- `y`: min(1,2,3) = 1
- `w`: not in second → ✘
✔ G.C.F. = xy
---
5. `mno`, `m²n²`, `n²o²m`
Write as: `m n o`, `m² n²`, `m n² o²`
- All have `m` and `n`
- `m`: min(1,2,1) = 1
- `n`: min(1,2,2) = 1
- `o`: not in second → ✘
✔ G.C.F. = mn
---
6. `a²b²m²`, `m²n²`, `b²n²`
Break down:
- `a²b²m²` → a² b² m²
- `m²n²` → m² n²
- `b²n²` → b² n²
Common variables? None appear in all three!
- `a` → only in first
- `b` → in first and third, not second
- `m` → in first and second, not third
- `n` → in second and third, not first
✔ No common variable → G.C.F. = 1
---
7. `c³b³a³`, `abc`, `a²b²c²`
All have `a`, `b`, `c`
- `a`: min(3,1,2) = 1
- `b`: min(3,1,2) = 1
- `c`: min(3,1,2) = 1
✔ G.C.F. = abc
---
8. `x³y³z³`, `xyz`, `x²y²z²`
All have `x`, `y`, `z`
- `x`: min(3,1,2) = 1
- `y`: min(3,1,2) = 1
- `z`: min(3,1,2) = 1
✔ G.C.F. = xyz
---
## 🔢 RIGHT COLUMN
---
1. `b²ac`, `b²c`, `a²b`
→ `a b² c`, `b² c`, `a² b`
Common variables? Only `b` appears in all.
- `b`: min(2,2,1) = 1
✔ G.C.F. = b
---
2. `m²n²z²`, `n²m²y²`, `mny`
→ All have `m` and `n`
- `m`: min(2,2,1) = 1
- `n`: min(2,2,1) = 1
- `z`, `y` → not in all
✔ G.C.F. = mn
---
3. `prt`, `pro`, `pqr`
→ `p r t`, `p r o`, `p q r`
Common variables? `p` and `r` are in all.
- `p`: min(1,1,1) = 1
- `r`: min(1,1,1) = 1
✔ G.C.F. = pr
---
4. `abc`, `b²c²`, `bca³`
→ `a b c`, `b² c²`, `a³ b c`
All have `b` and `c`
- `b`: min(1,2,1) = 1
- `c`: min(1,2,1) = 1
- `a`: not in second → ✘
✔ G.C.F. = bc
---
5. `xzo`, `x²z²`, `z²x²m`
→ `x z o`, `x² z²`, `x² z² m`
Common? Only `x` and `z`
- `x`: min(1,2,2) = 1
- `z`: min(1,2,2) = 1
✔ G.C.F. = xz
---
6. `t²u²v²`, `u²v²`, `u²v²`
All have `u²v²`
- `u`: min(2,2,2) = 2
- `v`: min(2,2,2) = 2
- `t`: only in first → ✘
✔ G.C.F. = u²v²
---
7. `m³n³a³`, `amn`, `m²n²a²`
All have `m`, `n`, `a`
- `m`: min(3,1,2) = 1
- `n`: min(3,1,2) = 1
- `a`: min(3,1,2) = 1
✔ G.C.F. = amn
---
8. `a³b³c³`, `abc`, `a²b²c²`
All have `a`, `b`, `c`
- `a`: min(3,1,2) = 1
- `b`: min(3,1,2) = 1
- `c`: min(3,1,2) = 1
✔ G.C.F. = abc
---
## 📝 FINAL ANSWERS:
Left Column:
1. c
2. q
3. r
4. xy
5. mn
6. 1
7. abc
8. xyz
Right Column:
1. b
2. mn
3. pr
4. bc
5. xz
6. u²v²
7. amn
8. abc
---
✔ Note: In problem #1 left column, you wrote “_a” — that’s incorrect. The correct G.C.F. is c.
Let me know if you want this formatted as a worksheet or need explanations for any specific one!
Parent Tip: Review the logic above to help your child master the concept of greatest common factor monomials worksheet.
