Polynomial Division

Grade 9 · algebra · 100 practice problems · read aloud

🔊 Listen to this explanation

Polynomial Division: Breaking Down Big Expressions

Polynomial division is like long division, but with variables! It's used to simplify complex polynomial expressions, find factors, and solve higher-degree equations. Mastering this helps you understand how polynomials behave.

🔍 How to Divide Polynomials (Long Division)

  1. Set it up: Write it like old-school long division. Dividend ÷ Divisor.
  2. Divide the first terms: Divide the first term of the dividend by the first term of the divisor.
  3. Multiply & subtract: Multiply your answer by the entire divisor, write it below, and subtract.
  4. Bring down & repeat: Bring down the next term and repeat the process until you can't bring down anymore.
  5. State your answer: Your final answer is the quotient plus the remainder over the divisor.

📚 Worked Examples

Example 1: Divide (x² + 5x + 6) by (x + 2)

  1. Divide: x² ÷ x = x
  2. Multiply: x(x + 2) = x² + 2x
  3. Subtract: (x² + 5x) - (x² + 2x) = 3x
  4. Bring down: +6 → 3x + 6
  5. Repeat: 3x ÷ x = 3 → 3(x + 2) = 3x + 6 → Subtract to get 0.

Answer: x + 3

Example 2: Divide (2x³ - 3x² + 4x - 5) by (x - 1)

  1. 2x³ ÷ x = 2x² → Multiply: 2x²(x - 1) = 2x³ - 2x²
  2. Subtract: (-3x² - (-2x²)) = -x². Bring down +4x.
  3. -x² ÷ x = -x → Multiply: -x(x - 1) = -x² + x
  4. Subtract: (4x - x) = 3x. Bring down -5.
  5. 3x ÷ x = 3 → Multiply: 3(x - 1) = 3x - 3 → Subtract: (-5 - (-3)) = -2.

Answer: 2x² - x + 3 + (-2)/(x - 1)

⚠️ Common Mistakes to Avoid

  • Missing terms: Always write polynomials in standard form. If a degree is missing (like no x²), add a 0x² as a placeholder!
  • Sign errors: Be super careful when subtracting the entire line. Use parentheses! -(x² + 2x) = -x² - 2x.
  • Forgetting the remainder: If you don't get zero at the end, write the remainder as a fraction over the divisor.

💡 Tips & Tricks

  • Check your work: Multiply your quotient by the divisor and add the remainder. You should get the original dividend back!
  • Spot the shortcuts: If the divisor is (x - c), look up "synthetic division" for a faster method.
  • Acronym: Remember the steps with Divide, Multiply, Subtract, Bring down (DMSB).

🎯 How to Practice

  • Start with simple divisors like (x + a) or (x - a).
  • Practice with divisors that result in a remainder.
  • Create your own problems and solve them, then check with multiplication.
  • Use online practice sites like Khan Academy for instant feedback.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 (12x³y² - 8x²y³ + 4xy⁴) ÷ (4xy) = ?

Hint: Divide each term in the polynomial separately by the monomial, applying the quotient rule for exponents.

Show the answer

Answer: 3x²y - 2xy² + y³

  1. Write the division as separate fractions** We can split the numerator into three terms, each divided by 4xy: = (12x³y²)/(4xy) - (8x²y³)/(4xy) + (4xy⁴)/(4xy) --- **
  2. Simplify coefficients** Divide the numbers first: 12 ÷ 4 = 3 8 ÷ 4 = 2 4 ÷ 4 = 1 So we have: = (3x³y²)/(xy) - (2x²y³)/(xy) + (1xy⁴)/(xy) --- **
  3. Simplify x terms** For the first term: (x³)/(x) = x^(3-1) = x² For the second term: (x²)/(x) = x^(2-1) = x¹ = x For the third term: (x)/(x) = x^(1-1) = x⁰ = 1 (so x disappears) So: = 3x²y²/y - 2xy³/y + 1y⁴/y --- **
  4. Simplify y terms** For the first term: (y²)/(y) = y^(2-1) = y¹ = y For the second term: (y³)/(y) = y^(3-1) = y² For the third term: (y⁴)/(y) = y^(4-1) = y³ So: = 3x²y - 2xy² + y³ --- **Final Answer:** 3x²y - 2xy² + y³

Let's solve step by step. We are dividing: (12x³y² - 8x²y³ + 4xy⁴) ÷ (4xy) --- **

2 (12x³y² - 18x²y⁴ + 6xy³) ÷ (3xy) = ?

Hint: Divide each term in the polynomial by the monomial separately, applying the quotient rule for exponents.

Show the answer

Answer: 4x²y - 6xy³ + 2y²

  1. Write the division as separate fractions** We can split the numerator into three terms, each divided by 3xy: = (12x³y²)/(3xy) - (18x²y⁴)/(3xy) + (6xy³)/(3xy) --- **
  2. Simplify coefficients** 12 ÷ 3 = 4 18 ÷ 3 = 6 6 ÷ 3 = 2 So we have: 4 * (x³y²)/(xy) - 6 * (x²y⁴)/(xy) + 2 * (xy³)/(xy) --- **
  3. Simplify x terms** For (x³y²)/(xy): x³ / x¹ = x^(3-1) = x² For (x²y⁴)/(xy): x² / x¹ = x^(2-1) = x¹ = x For (xy³)/(xy): x¹ / x¹ = x^(1-1) = x⁰ = 1 So x terms simplify to: Term 1: x² Term 2: x Term 3: 1 --- **
  4. Simplify y terms** For (x³y²)/(xy): y² / y¹ = y^(2-1) = y¹ = y For (x²y⁴)/(xy): y⁴ / y¹ = y^(4-1) = y³ For (xy³)/(xy): y³ / y¹ = y^(3-1) = y² --- **
  5. Combine results** Term 1: 4 * x² * y = 4x²y Term 2: -6 * x * y³ = -6xy³ Term 3: +2 * 1 * y² = 2y² So the result is: 4x²y - 6xy³ + 2y² --- **Final answer:** 4x²y - 6xy³ + 2y²

Let's solve step by step. We are dividing: (12x³y² - 18x²y⁴ + 6xy³) ÷ (3xy) --- **

3 (12x³y² - 18x²y⁴ + 24xy³) ÷ (6xy) = ?

Hint: Divide each term in the polynomial separately by the monomial, applying the quotient rule for exponents where necessary.

Show the answer

Answer: 2x²y - 3xy³ + 4y²

  1. Write the division as separate fractions for each term in the numerator: = (12x³y²)/(6xy) - (18x²y⁴)/(6xy) + (24xy³)/(6xy)
  2. Simplify each fraction separately by dividing coefficients and subtracting exponents (since x^a / x^b = x^(a-b) and same for y): First term: (12x³y²)/(6xy) - Divide coefficients: 12/6 = 2 - For x: x³/x¹ = x^(3-1) = x² - For y: y²/y¹ = y^(2-1) = y¹ = y So first term = 2x²y Second term: (18x²y⁴)/(6xy) - Divide coefficients: 18/6 = 3 - For x: x²/x¹ = x^(2-1) = x¹ = x - For y: y⁴/y¹ = y^(4-1) = y³ So second term = 3xy³ (with a minus sign from the original expression) Third term: (24xy³)/(6xy) - Divide coefficients: 24/6 = 4 - For x: x¹/x¹ = x^(1-1) = x⁰ = 1 (so x disappears) - For y: y³/y¹ = y^(3-1) = y² So third term = 4y²
  3. Combine all terms with their signs: = 2x²y - 3xy³ + 4y² Final answer: 2x²y - 3xy³ + 4y²

Let's solve step by step: We are dividing: (12x³y² - 18x²y⁴ + 24xy³) ÷ (6xy)

4 (15x⁴y³ - 25x³y² + 35x²y) ÷ (5x²y) = ?

Hint: Divide each term in the numerator by the monomial denominator separately, applying exponent rules for division

Show the answer

Answer: 3x²y² - 5xy + 7

  1. Write the division as separate fractions: (15x⁴y³)/(5x²y) - (25x³y²)/(5x²y) + (35x²y)/(5x²y)
  2. Divide coefficients: 15 ÷ 5 = 3, 25 ÷ 5 = 5, 35 ÷ 5 = 7
  3. Apply exponent rules for x terms: x⁴ ÷ x² = x², x³ ÷ x² = x¹, x² ÷ x² = x⁰ = 1
  4. Apply exponent rules for y terms: y³ ÷ y¹ = y², y² ÷ y¹ = y¹, y¹ ÷ y¹ = y⁰ = 1
  5. Combine results: 3x²y² - 5xy + 7
  6. Final answer: 3x²y² - 5xy + 7

5 (15x⁴y³ - 25x³y² + 10x²y) ÷ (5x²y) = ?

Hint: Divide each term in the numerator by the monomial denominator separately, applying exponent rules for division

Show the answer

Answer: 3x²y² - 5xy + 2

  1. Divide each term in the polynomial by the monomial (5x²y)
  2. First term: (15x⁴y³) ÷ (5x²y) = (15÷5)x^(4-2)y^(3-1) = 3x²y²
  3. Second term: (-25x³y²) ÷ (5x²y) = (-25÷5)x^(3-2)y^(2-1) = -5xy
  4. Third term: (10x²y) ÷ (5x²y) = (10÷5)x^(2-2)y^(1-1) = 2
  5. Combine the results: 3x²y² - 5xy + 2

The answer is 3x²y² - 5xy + 2.

6 (15x⁴y³ - 25x³y² + 35x²y⁴) ÷ (5x²y) = ?

Hint: Divide each term in the numerator by the monomial denominator separately, applying exponent rules for division

Show the answer

Answer: 3x²y² - 5xy + 7y³

  1. Write the division as separate fractions: (15x⁴y³)/(5x²y) - (25x³y²)/(5x²y) + (35x²y⁴)/(5x²y)
  2. Divide coefficients: 15÷5=3, 25÷5=5, 35÷5=7
  3. Apply exponent rules for x terms: x⁴÷x²=x², x³÷x²=x¹, x²÷x²=x⁰=1
  4. Apply exponent rules for y terms: y³÷y=y², y²÷y=y¹, y⁴÷y=y³
  5. Combine results: 3x²y² - 5x¹y¹ + 7x⁰y³
  6. Simplify: 3x²y² - 5xy + 7y³

The answer is 3x²y² - 5xy + 7y³.

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