Algebra 1 Worksheets | Monomials and Polynomials Worksheets - Free Printable
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Step-by-step solution for: Algebra 1 Worksheets | Monomials and Polynomials Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Algebra 1 Worksheets | Monomials and Polynomials Worksheets
Let’s solve each polynomial division problem one by one. We’ll use long division or synthetic division where appropriate, and remember to write any remainder as a fraction over the divisor.
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1) (-3y³ + 16y² + 3y - 10) ÷ (y - 3)
We’ll use synthetic division with root = 3.
Coefficients: -3, 16, 3, -10
Bring down -3
Multiply by 3 → -9 → add to 16 → 7
Multiply by 3 → 21 → add to 3 → 24
Multiply by 3 → 72 → add to -10 → 62
So quotient is: -3y² + 7y + 24, remainder 62
Answer: -3y² + 7y + 24 + 62/(y - 3)
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2) (-d² - 17) ÷ (d + 8)
Rewrite dividend: -d² + 0d - 17
Synthetic division with root = -8
Coefficients: -1, 0, -17
Bring down -1
Multiply by -8 → 8 → add to 0 → 8
Multiply by -8 → -64 → add to -17 → -81
Quotient: -d + 8, remainder -81
Answer: -d + 8 - 81/(d + 8)
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3) (-4y² + 20y - 18) ÷ (y + 9)
Root = -9
Coefficients: -4, 20, -18
Bring down -4
Multiply by -9 → 36 → add to 20 → 56
Multiply by -9 → -504 → add to -18 → -522
Quotient: -4y + 56, remainder -522
Answer: -4y + 56 - 522/(y + 9)
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4) (-3n² - 10n - 17) ÷ (n + 3)
Root = -3
Coefficients: -3, -10, -17
Bring down -3
Multiply by -3 → 9 → add to -10 → -1
Multiply by -3 → 3 → add to -17 → -14
Quotient: -3n -1, remainder -14
Answer: -3n - 1 - 14/(n + 3)
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5) (b² - 20b + 5) ÷ (b - 9)
Root = 9
Coefficients: 1, -20, 5
Bring down 1
Multiply by 9 → 9 → add to -20 → -11
Multiply by 9 → -99 → add to 5 → -94
Quotient: b - 11, remainder -94
Answer: b - 11 - 94/(b - 9)
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6) (-2d² + 17) ÷ (d + 5)
Rewrite: -2d² + 0d + 17
Root = -5
Coefficients: -2, 0, 17
Bring down -2
Multiply by -5 → 10 → add to 0 → 10
Multiply by -5 → -50 → add to 17 → -33
Quotient: -2d + 10, remainder -33
Answer: -2d + 10 - 33/(d + 5)
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7) (-r² - 12) ÷ (r + 2)
Rewrite: -r² + 0r - 12
Root = -2
Coefficients: -1, 0, -12
Bring down -1
Multiply by -2 → 2 → add to 0 → 2
Multiply by -2 → -4 → add to -12 → -16
Quotient: -r + 2, remainder -16
Answer: -r + 2 - 16/(r + 2)
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8) (3x³ - 13x² + 7x - 10) ÷ (x - 3)
Root = 3
Coefficients: 3, -13, 7, -10
Bring down 3
Multiply by 3 → 9 → add to -13 → -4
Multiply by 3 → -12 → add to 7 → -5
Multiply by 3 → -15 → add to -10 → -25
Quotient: 3x² - 4x - 5, remainder -25
Answer: 3x² - 4x - 5 - 25/(x - 3)
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9) (b² - 16) ÷ (b + 3)
Note: b² - 16 = (b - 4)(b + 4), but we’re dividing by (b + 3), so no factoring help.
Use synthetic division with root = -3
Coefficients: 1, 0, -16
Bring down 1
Multiply by -3 → -3 → add to 0 → -3
Multiply by -3 → 9 → add to -16 → -7
Quotient: b - 3, remainder -7
Answer: b - 3 - 7/(b + 3)
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10) (-4n³ + 8n² - 19n + 8) ÷ (n - 7)
Root = 7
Coefficients: -4, 8, -19, 8
Bring down -4
Multiply by 7 → -28 → add to 8 → -20
Multiply by 7 → -140 → add to -19 → -159
Multiply by 7 → -1113 → add to 8 → -1105
Quotient: -4n² -20n -159, remainder -1105
Answer: -4n² - 20n - 159 - 1105/(n - 7)
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Final Answer:
1) -3y² + 7y + 24 + 62/(y - 3)
2) -d + 8 - 81/(d + 8)
3) -4y + 56 - 522/(y + 9)
4) -3n - 1 - 14/(n + 3)
5) b - 11 - 94/(b - 9)
6) -2d + 10 - 33/(d + 5)
7) -r + 2 - 16/(r + 2)
8) 3x² - 4x - 5 - 25/(x - 3)
9) b - 3 - 7/(b + 3)
10) -4n² - 20n - 159 - 1105/(n - 7)
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1) (-3y³ + 16y² + 3y - 10) ÷ (y - 3)
We’ll use synthetic division with root = 3.
Coefficients: -3, 16, 3, -10
Bring down -3
Multiply by 3 → -9 → add to 16 → 7
Multiply by 3 → 21 → add to 3 → 24
Multiply by 3 → 72 → add to -10 → 62
So quotient is: -3y² + 7y + 24, remainder 62
Answer: -3y² + 7y + 24 + 62/(y - 3)
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2) (-d² - 17) ÷ (d + 8)
Rewrite dividend: -d² + 0d - 17
Synthetic division with root = -8
Coefficients: -1, 0, -17
Bring down -1
Multiply by -8 → 8 → add to 0 → 8
Multiply by -8 → -64 → add to -17 → -81
Quotient: -d + 8, remainder -81
Answer: -d + 8 - 81/(d + 8)
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3) (-4y² + 20y - 18) ÷ (y + 9)
Root = -9
Coefficients: -4, 20, -18
Bring down -4
Multiply by -9 → 36 → add to 20 → 56
Multiply by -9 → -504 → add to -18 → -522
Quotient: -4y + 56, remainder -522
Answer: -4y + 56 - 522/(y + 9)
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4) (-3n² - 10n - 17) ÷ (n + 3)
Root = -3
Coefficients: -3, -10, -17
Bring down -3
Multiply by -3 → 9 → add to -10 → -1
Multiply by -3 → 3 → add to -17 → -14
Quotient: -3n -1, remainder -14
Answer: -3n - 1 - 14/(n + 3)
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5) (b² - 20b + 5) ÷ (b - 9)
Root = 9
Coefficients: 1, -20, 5
Bring down 1
Multiply by 9 → 9 → add to -20 → -11
Multiply by 9 → -99 → add to 5 → -94
Quotient: b - 11, remainder -94
Answer: b - 11 - 94/(b - 9)
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6) (-2d² + 17) ÷ (d + 5)
Rewrite: -2d² + 0d + 17
Root = -5
Coefficients: -2, 0, 17
Bring down -2
Multiply by -5 → 10 → add to 0 → 10
Multiply by -5 → -50 → add to 17 → -33
Quotient: -2d + 10, remainder -33
Answer: -2d + 10 - 33/(d + 5)
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7) (-r² - 12) ÷ (r + 2)
Rewrite: -r² + 0r - 12
Root = -2
Coefficients: -1, 0, -12
Bring down -1
Multiply by -2 → 2 → add to 0 → 2
Multiply by -2 → -4 → add to -12 → -16
Quotient: -r + 2, remainder -16
Answer: -r + 2 - 16/(r + 2)
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8) (3x³ - 13x² + 7x - 10) ÷ (x - 3)
Root = 3
Coefficients: 3, -13, 7, -10
Bring down 3
Multiply by 3 → 9 → add to -13 → -4
Multiply by 3 → -12 → add to 7 → -5
Multiply by 3 → -15 → add to -10 → -25
Quotient: 3x² - 4x - 5, remainder -25
Answer: 3x² - 4x - 5 - 25/(x - 3)
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9) (b² - 16) ÷ (b + 3)
Note: b² - 16 = (b - 4)(b + 4), but we’re dividing by (b + 3), so no factoring help.
Use synthetic division with root = -3
Coefficients: 1, 0, -16
Bring down 1
Multiply by -3 → -3 → add to 0 → -3
Multiply by -3 → 9 → add to -16 → -7
Quotient: b - 3, remainder -7
Answer: b - 3 - 7/(b + 3)
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10) (-4n³ + 8n² - 19n + 8) ÷ (n - 7)
Root = 7
Coefficients: -4, 8, -19, 8
Bring down -4
Multiply by 7 → -28 → add to 8 → -20
Multiply by 7 → -140 → add to -19 → -159
Multiply by 7 → -1113 → add to 8 → -1105
Quotient: -4n² -20n -159, remainder -1105
Answer: -4n² - 20n - 159 - 1105/(n - 7)
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Final Answer:
1) -3y² + 7y + 24 + 62/(y - 3)
2) -d + 8 - 81/(d + 8)
3) -4y + 56 - 522/(y + 9)
4) -3n - 1 - 14/(n + 3)
5) b - 11 - 94/(b - 9)
6) -2d + 10 - 33/(d + 5)
7) -r + 2 - 16/(r + 2)
8) 3x² - 4x - 5 - 25/(x - 3)
9) b - 3 - 7/(b + 3)
10) -4n² - 20n - 159 - 1105/(n - 7)
Parent Tip: Review the logic above to help your child master the concept of multiplying dividing polynomials worksheet.
