Performing Operations on Polynomials 2) + (6v - Free Printable
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Step-by-step solution for: Performing Operations on Polynomials 2) + (6v
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Show Answer Key & Explanations
Step-by-step solution for: Performing Operations on Polynomials 2) + (6v
Explanation:
Let's solve each problem step by step. We will simplify the expressions by combining like terms and using the distributive property where needed.
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1) (3x⁴ − 2 + 8x²) + (6x² + 4 − 7x⁴)
Step 1: Remove parentheses.
= 3x⁴ − 2 + 8x² + 6x² + 4 − 7x⁴
Step 2: Group like terms.
= (3x⁴ − 7x⁴) + (8x² + 6x²) + (−2 + 4)
= −4x⁴ + 14x² + 2
---
2) (4a² − 5 − v²) − (3 − 6v² − 3v²)
Step 1: Remove parentheses. Remember to distribute the minus sign.
= 4a² − 5 − v² − 3 + 6v² + 3v²
Step 2: Group like terms.
= 4a² + (−v² + 6v² + 3v²) + (−5 − 3)
= 4a² + 8v² − 8
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3) (2x⁴ + x) + (7x − 2x²) − (4x − 6x⁴ + 5x²)
Step 1: Remove parentheses.
= 2x⁴ + x + 7x − 2x² − 4x + 6x⁴ − 5x²
Step 2: Group like terms.
= (2x⁴ + 6x⁴) + (−2x² − 5x²) + (x + 7x − 4x)
= 8x⁴ − 7x² + 4x
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4) 3x(x² − 7x + 2)
Step 1: Distribute 3x to each term inside the parentheses.
= 3x · x² + 3x · (−7x) + 3x · 2
= 3x³ − 21x² + 6x
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5) (6a + 5)(3a − 5)
Step 1: Use the distributive property (FOIL).
= 6a · 3a + 6a · (−5) + 5 · 3a + 5 · (−5)
= 18a² − 30a + 15a − 25
= 18a² − 15a − 25
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6) (8a − 2)(3a² + 2a − 2)
Step 1: Distribute each term in the first parentheses.
= 8a(3a² + 2a − 2) − 2(3a² + 2a − 2)
= 24a³ + 16a² − 16a − 6a² − 4a + 4
Step 2: Combine like terms.
= 24a³ + (16a² − 6a²) + (−16a − 4a) + 4
= 24a³ + 10a² − 20a + 4
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7) (3a + 2b)(8a + 8b)
Step 1: Use FOIL.
= 3a · 8a + 3a · 8b + 2b · 8a + 2b · 8b
= 24a² + 24ab + 16ab + 16b²
= 24a² + 40ab + 16b²
---
8) (8m² − 2mn − 7n²)(6m + 3n)
Step 1: Distribute each term.
= 8m²(6m + 3n) − 2mn(6m + 3n) − 7n²(6m + 3n)
= 48m³ + 24m²n − 12m²n − 6mn² − 42mn² − 21n³
Step 2: Combine like terms.
= 48m³ + (24m²n − 12m²n) + (−6mn² − 42mn²) − 21n³
= 48m³ + 12m²n − 48mn² − 21n³
---
9) (8b − 7)(8b + 7)
This is a difference of squares: (a − b)(a + b) = a² − b²
= (8b)² − (7)² = 64b² − 49
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10) (2y + x)(2y − x)
This is also a difference of squares.
= (2y)² − (x)² = 4y² − x²
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11) (5p − 3)²
This is a square of a binomial: (a − b)² = a² − 2ab + b²
= (5p)² − 2(5p)(3) + (3)² = 25p² − 30p + 9
---
12) (6x + y)²
Use (a + b)² = a² + 2ab + b²
= (6x)² + 2(6x)(y) + y² = 36x² + 12xy + y²
---
13) (5k³ + 5k⁵ + 40k⁴) ÷ 10k²
Step 1: Divide each term by 10k².
= (5k³)/(10k²) + (5k⁵)/(10k²) + (40k⁴)/(10k²)
= (1/2)k + (1/2)k³ + 4k²
---
14) (3k²m² + 3k²m² + 2km) / 8km
Step 1: Combine like terms in the numerator.
= (6k²m² + 2km) / 8km
Step 2: Divide each term by 8km.
= (6k²m²)/(8km) + (2km)/(8km)
= (3/4)km + 1/4
---
15) (m² − m − 80) ÷ (m − 9)
We can try polynomial division or see if it factors.
But m² − m − 80 does not factor easily with (m − 9).
Let’s do long division.
Divide m² − m − 80 by m − 9.
m² ÷ m = m
Multiply: m(m − 9) = m² − 9m
Subtract: (m² − m − 80) − (m² − 9m) = 8m − 80
8m ÷ m = 8
Multiply: 8(m − 9) = 8m − 72
Subtract: (8m − 80) − (8m − 72) = −8
So the result is: m + 8 − 8/(m − 9)
But since this is likely for simplification, and the division is not exact, we write:
= m + 8 − 8/(m − 9)
But let's check if it factors.
m² − m − 80: discriminant = 1 + 320 = 321 → not a perfect square.
So the division gives:
Quotient: m + 8, remainder: −8
So: (m² − m − 80)/(m − 9) = m + 8 − 8/(m − 9)
But maybe the problem expects a simplified expression, so we can write:
= m + 8 − 8/(m − 9)
But let's double-check the problem.
Wait, maybe it's a typo or I misread.
But let's proceed.
---
16) (n³ + 10n² + 29n + 60) ÷ (n + 7)
Use polynomial division.
n³ ÷ n = n²
n²(n + 7) = n³ + 7n²
Subtract: (n³ + 10n² + 29n + 60) − (n³ + 7n²) = 3n² + 29n + 60
3n² ÷ n = 3n
3n(n + 7) = 3n² + 21n
Subtract: (3n² + 29n + 60) − (3n² + 21n) = 8n + 60
8n ÷ n = 8
8(n + 7) = 8n + 56
Subtract: (8n + 60) − (8n + 56) = 4
So: quotient = n² + 3n + 8, remainder = 4
Answer: n² + 3n + 8 + 4/(n + 7)
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17) (8v² − v − 4) ÷ (8v − 9)
Divide 8v² − v − 4 by 8v − 9.
8v² ÷ 8v = v
v(8v − 9) = 8v² − 9v
Subtract: (8v² − v − 4) − (8v² − 9v) = 8v − 4
8v ÷ 8v = 1
1(8v − 9) = 8v − 9
Subtract: (8v − 4) − (8v − 9) = 5
So: quotient = v + 1, remainder = 5
Answer: v + 1 + 5/(8v − 9)
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18) (2a² + 13a + 21) ÷ (2a + 7)
Try factoring numerator.
2a² + 13a + 21
Find two numbers that multiply to 2×21 = 42, add to 13 → 6 and 7
So: 2a² + 6a + 7a + 21 = 2a(a + 3) + 7(a + 3) = (2a + 7)(a + 3)
So: (2a + 7)(a + 3) ÷ (2a + 7) = a + 3
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Final Answer:
1) −4x⁴ + 14x² + 2
2) 4a² + 8v² − 8
3) 8x⁴ − 7x² + 4x
4) 3x³ − 21x² + 6x
5) 18a² − 15a − 25
6) 24a³ + 10a² − 20a + 4
7) 24a² + 40ab + 16b²
8) 48m³ + 12m²n − 48mn² − 21n³
9) 64b² − 49
10) 4y² − x²
11) 25p² − 30p + 9
12) 36x² + 12xy + y²
13) (1/2)k + (1/2)k³ + 4k²
14) (3/4)km + 1/4
15) m + 8 − 8/(m − 9)
16) n² + 3n + 8 + 4/(n + 7)
17) v + 1 + 5/(8v − 9)
18) a + 3
Let's solve each problem step by step. We will simplify the expressions by combining like terms and using the distributive property where needed.
---
1) (3x⁴ − 2 + 8x²) + (6x² + 4 − 7x⁴)
Step 1: Remove parentheses.
= 3x⁴ − 2 + 8x² + 6x² + 4 − 7x⁴
Step 2: Group like terms.
= (3x⁴ − 7x⁴) + (8x² + 6x²) + (−2 + 4)
= −4x⁴ + 14x² + 2
---
2) (4a² − 5 − v²) − (3 − 6v² − 3v²)
Step 1: Remove parentheses. Remember to distribute the minus sign.
= 4a² − 5 − v² − 3 + 6v² + 3v²
Step 2: Group like terms.
= 4a² + (−v² + 6v² + 3v²) + (−5 − 3)
= 4a² + 8v² − 8
---
3) (2x⁴ + x) + (7x − 2x²) − (4x − 6x⁴ + 5x²)
Step 1: Remove parentheses.
= 2x⁴ + x + 7x − 2x² − 4x + 6x⁴ − 5x²
Step 2: Group like terms.
= (2x⁴ + 6x⁴) + (−2x² − 5x²) + (x + 7x − 4x)
= 8x⁴ − 7x² + 4x
---
4) 3x(x² − 7x + 2)
Step 1: Distribute 3x to each term inside the parentheses.
= 3x · x² + 3x · (−7x) + 3x · 2
= 3x³ − 21x² + 6x
---
5) (6a + 5)(3a − 5)
Step 1: Use the distributive property (FOIL).
= 6a · 3a + 6a · (−5) + 5 · 3a + 5 · (−5)
= 18a² − 30a + 15a − 25
= 18a² − 15a − 25
---
6) (8a − 2)(3a² + 2a − 2)
Step 1: Distribute each term in the first parentheses.
= 8a(3a² + 2a − 2) − 2(3a² + 2a − 2)
= 24a³ + 16a² − 16a − 6a² − 4a + 4
Step 2: Combine like terms.
= 24a³ + (16a² − 6a²) + (−16a − 4a) + 4
= 24a³ + 10a² − 20a + 4
---
7) (3a + 2b)(8a + 8b)
Step 1: Use FOIL.
= 3a · 8a + 3a · 8b + 2b · 8a + 2b · 8b
= 24a² + 24ab + 16ab + 16b²
= 24a² + 40ab + 16b²
---
8) (8m² − 2mn − 7n²)(6m + 3n)
Step 1: Distribute each term.
= 8m²(6m + 3n) − 2mn(6m + 3n) − 7n²(6m + 3n)
= 48m³ + 24m²n − 12m²n − 6mn² − 42mn² − 21n³
Step 2: Combine like terms.
= 48m³ + (24m²n − 12m²n) + (−6mn² − 42mn²) − 21n³
= 48m³ + 12m²n − 48mn² − 21n³
---
9) (8b − 7)(8b + 7)
This is a difference of squares: (a − b)(a + b) = a² − b²
= (8b)² − (7)² = 64b² − 49
---
10) (2y + x)(2y − x)
This is also a difference of squares.
= (2y)² − (x)² = 4y² − x²
---
11) (5p − 3)²
This is a square of a binomial: (a − b)² = a² − 2ab + b²
= (5p)² − 2(5p)(3) + (3)² = 25p² − 30p + 9
---
12) (6x + y)²
Use (a + b)² = a² + 2ab + b²
= (6x)² + 2(6x)(y) + y² = 36x² + 12xy + y²
---
13) (5k³ + 5k⁵ + 40k⁴) ÷ 10k²
Step 1: Divide each term by 10k².
= (5k³)/(10k²) + (5k⁵)/(10k²) + (40k⁴)/(10k²)
= (1/2)k + (1/2)k³ + 4k²
---
14) (3k²m² + 3k²m² + 2km) / 8km
Step 1: Combine like terms in the numerator.
= (6k²m² + 2km) / 8km
Step 2: Divide each term by 8km.
= (6k²m²)/(8km) + (2km)/(8km)
= (3/4)km + 1/4
---
15) (m² − m − 80) ÷ (m − 9)
We can try polynomial division or see if it factors.
But m² − m − 80 does not factor easily with (m − 9).
Let’s do long division.
Divide m² − m − 80 by m − 9.
m² ÷ m = m
Multiply: m(m − 9) = m² − 9m
Subtract: (m² − m − 80) − (m² − 9m) = 8m − 80
8m ÷ m = 8
Multiply: 8(m − 9) = 8m − 72
Subtract: (8m − 80) − (8m − 72) = −8
So the result is: m + 8 − 8/(m − 9)
But since this is likely for simplification, and the division is not exact, we write:
= m + 8 − 8/(m − 9)
But let's check if it factors.
m² − m − 80: discriminant = 1 + 320 = 321 → not a perfect square.
So the division gives:
Quotient: m + 8, remainder: −8
So: (m² − m − 80)/(m − 9) = m + 8 − 8/(m − 9)
But maybe the problem expects a simplified expression, so we can write:
= m + 8 − 8/(m − 9)
But let's double-check the problem.
Wait, maybe it's a typo or I misread.
But let's proceed.
---
16) (n³ + 10n² + 29n + 60) ÷ (n + 7)
Use polynomial division.
n³ ÷ n = n²
n²(n + 7) = n³ + 7n²
Subtract: (n³ + 10n² + 29n + 60) − (n³ + 7n²) = 3n² + 29n + 60
3n² ÷ n = 3n
3n(n + 7) = 3n² + 21n
Subtract: (3n² + 29n + 60) − (3n² + 21n) = 8n + 60
8n ÷ n = 8
8(n + 7) = 8n + 56
Subtract: (8n + 60) − (8n + 56) = 4
So: quotient = n² + 3n + 8, remainder = 4
Answer: n² + 3n + 8 + 4/(n + 7)
---
17) (8v² − v − 4) ÷ (8v − 9)
Divide 8v² − v − 4 by 8v − 9.
8v² ÷ 8v = v
v(8v − 9) = 8v² − 9v
Subtract: (8v² − v − 4) − (8v² − 9v) = 8v − 4
8v ÷ 8v = 1
1(8v − 9) = 8v − 9
Subtract: (8v − 4) − (8v − 9) = 5
So: quotient = v + 1, remainder = 5
Answer: v + 1 + 5/(8v − 9)
---
18) (2a² + 13a + 21) ÷ (2a + 7)
Try factoring numerator.
2a² + 13a + 21
Find two numbers that multiply to 2×21 = 42, add to 13 → 6 and 7
So: 2a² + 6a + 7a + 21 = 2a(a + 3) + 7(a + 3) = (2a + 7)(a + 3)
So: (2a + 7)(a + 3) ÷ (2a + 7) = a + 3
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Final Answer:
1) −4x⁴ + 14x² + 2
2) 4a² + 8v² − 8
3) 8x⁴ − 7x² + 4x
4) 3x³ − 21x² + 6x
5) 18a² − 15a − 25
6) 24a³ + 10a² − 20a + 4
7) 24a² + 40ab + 16b²
8) 48m³ + 12m²n − 48mn² − 21n³
9) 64b² − 49
10) 4y² − x²
11) 25p² − 30p + 9
12) 36x² + 12xy + y²
13) (1/2)k + (1/2)k³ + 4k²
14) (3/4)km + 1/4
15) m + 8 − 8/(m − 9)
16) n² + 3n + 8 + 4/(n + 7)
17) v + 1 + 5/(8v − 9)
18) a + 3
Parent Tip: Review the logic above to help your child master the concept of operations with polynomials worksheet.
