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Learn Index · Patterns and Algebra Grade 11–12

Dividing polynomials

Dividing a polynomial by a monomial or a binomial.

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Dividing polynomials

Just as 47 ÷ 5 = 9 R2, polynomials divide with a quotient and a remainder:

P(x) = (divisor)(quotient) + remainder

Method 1: Long division

Example: (x³ + 2x² − 5x − 6) ÷ (x − 2)

  1. Divide: x³ ÷ x = x². Write x².
  2. Multiply: x²(x − 2) = x³ − 2x².
  3. Subtract: (x³ + 2x²) − (x³ − 2x²) = 4x². Bring down −5x.
  4. Repeat: 4x² ÷ x = 4x. 4x(x − 2) = 4x² − 8x. Subtract: 3x. Bring down −6.
  5. Repeat: 3x ÷ x = 3. 3(x − 2) = 3x − 6. Subtract: 0.

Quotient: x² + 4x + 3. Remainder: 0.

So x³ + 2x² − 5x − 6 = (x − 2)(x² + 4x + 3) = (x − 2)(x + 1)(x + 3).

Tip: Include missing terms with 0 coefficients. For x³ − 8, write x³ + 0x² + 0x − 8.

Method 2: Synthetic division

For divisors of the form x − k, use only the coefficients.

Example: (2x³ − 3x² + 4x − 5) ÷ (x − 1), so k = 1

2 −3 4 −5
k = 1 2 −1 3
result 2 −1 3 −2

Steps: bring down 2; multiply by k and add to the next coefficient; repeat.

Quotient: 2x² − x + 3. Remainder: −2.

For x + 3, use k = −3.

The remainder theorem

When P(x) is divided by x − k, the remainder is P(k).

Example: Remainder when P(x) = x³ − 4x + 1 is divided by x − 2?

P(2) = 8 − 8 + 1 = 1

No division needed!

The factor theorem

x − k is a factor of P(x) exactly when P(k) = 0.

Example: Is x + 1 a factor of x³ + 2x² − x − 2?

P(−1) = −1 + 2 + 1 − 2 = 0 → yes.

Factoring cubics

  1. Test integer factors of the constant term (±1, ±2, …) using the factor theorem.
  2. When P(k) = 0, divide by x − k.
  3. Factor the quadratic quotient.

Example: x³ − 7x + 6

  • Try 1: 1 − 7 + 6 = 0 ✓ → (x − 1) is a factor.
  • Divide: x² + x − 6 = (x + 3)(x − 2).
  • (x − 1)(x + 3)(x − 2)

Worked examples

Example 1: Remainder of (x² + 5x − 2) ÷ (x + 3)? P(−3) = 9 − 15 − 2 = −8.

Example 2: Is x − 3 a factor of x³ − 27? P(3) = 0 → yes.

Example 3: (x² − 9) ÷ (x − 3)? x + 3.

Common mistakes

  • Forgetting placeholder zeros for missing terms.
  • Using k = 2 for x + 2 (it should be −2).
  • Subtraction sign errors in long division.

Try it yourself

  1. Divide (x² + 7x + 10) ÷ (x + 2).
  2. Use synthetic division: (x³ − 6x² + 11x − 6) ÷ (x − 1).
  3. Find the remainder of (2x³ + x − 4) ÷ (x − 2).
  4. Factor x³ − 2x² − 5x + 6.

Answers: 1) x + 5 2) x² − 5x + 6 3) 14 4) (x − 1)(x + 2)(x − 3)

Quick recap

  • Use long division or synthetic division for polynomial division.
  • Remainder theorem: remainder = P(k).
  • Factor theorem: P(k) = 0 means x − k is a factor.