Polynomial Division (Long & Synthetic)

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Divide polynomials using long division.
  • Use synthetic division when the divisor has the form \((x - c)\).
  • Interpret quotients and remainders clearly.

Key Ideas

  • Polynomial long division works just like numeric long division—divide the leading terms, multiply back, subtract, and repeat.
  • Synthetic division is a shortcut method, but it only applies when dividing by a linear divisor of the form \((x - c)\).
  • Always fill in zero placeholders for missing powers before beginning division.

PLACEHOLDER: Insert long division layout figure here.

Common Problem Types

1. Long Division With Missing Terms

Fill in placeholders (e.g., \(3x^3 - 5x\) becomes \(3x^3 + 0x^2 - 5x\)).

2. Synthetic Division With \((x - c)\)

Use \(c\) (not \(-c\)) directly in the synthetic table.

3. Identifying Quotient + Remainder

Write remainder as a fraction over the divisor.

4. Dividing Higher-Degree Polynomials

Expect multi-step cycles of dividing, multiplying, and subtracting.

Strategies

  • Begin by rewriting both dividend and divisor in descending powers of \(x\).
  • In long division, focus only on leading terms when choosing the next part of the quotient.
  • In synthetic division, copy coefficients carefully—including zeros.
  • Remainders should always be written as \(\dfrac{\text{remainder}}{\text{divisor}}\).
  • If unsure which method to use, remember:
    • \((x - c)\) → synthetic
    • Anything else → long division

Worked Examples

Example 1 — Long Division

Divide:
\[ (3x^3 - 5x + 2) \div (x - 1) \]

Rewrite with a placeholder term: \[ 3x^3 + 0x^2 - 5x + 2 \]

PLACEHOLDER: Insert long division steps figure here.

Final answer: \[ 3x^2 + 3x - 2 + \frac{0}{x - 1} \]

Example 2 — Synthetic Division

Divide:
\[ 2x^3 + 3x^2 - 4x + 5 \quad \text{by} \quad (x + 2) \]

Since \((x + 2) = (x - (-2))\), use \(c = -2\).

PLACEHOLDER: Insert synthetic division table figure.

Resulting coefficients give the quotient: \[ 2x^2 - x - 2 \]

Remainder:
\[ 1 \]

Final answer: \[ 2x^2 - x - 2 + \frac{1}{x + 2} \]

WarningCommon Mistakes
  • Forgetting zero placeholders for missing powers of \(x\).
  • Using synthetic division with divisors that are not of the form \((x - c)\).
  • Mixing up the sign of \(c\) when setting up synthetic division.

Practice Problems

  1. \((x^2 + 5x + 6) \div (x + 2)\)
  2. \((2x^3 - x^2 + 4) \div (x - 1)\)
  3. \((4x^3 + 8x) \div (x + 2)\)
  4. \((3x^3 + 7x^2 - 2x + 8) \div (x - 3)\)
  5. \((x^4 - 1) \div (x - 1)\)

1.
\[ x + 3 \]

2.
\[ 2x^2 + x + 1 + \frac{5}{x - 1} \]

3.
\[ 4x^2 - 8x + 16 \]

4.
\[ 3x^2 + 16x + 46 + \frac{146}{x - 3} \]

5.
\[ x^3 + x^2 + x + 1 \]

Summary

  • Use long division for any divisor; use synthetic division only for \((x - c)\).
  • Fill in missing powers of \(x\) to avoid errors during division.
  • Divide leading terms first and work step-by-step.
  • Write remainders as fractions over the divisor.
  • Synthetic division provides a fast shortcut when applicable.
  • Always rewrite polynomials in descending order.
  • For synthetic division, use the number \(c\) from \((x - c)\).
  • Check your remainder: plugging \(c\) into the original polynomial should produce it.
  • Long division becomes much easier if you align like terms vertically.