Polynomial Division & Special Products

Fibo  /  Math  /  Algebra  /  Poly Division
Skill: polynomial-division

Polynomial Division & Special Products

Some multiplications are worth memorizing: (a + b)², (a − b)², and (a + b)(a − b) expand instantly once you see the pattern. Then learn to divide polynomials — by long division and by the shortcut of synthetic division — and meet the Remainder Theorem.

1 Understand

The core idea in plain language.

What it is. Three special products appear everywhere: (a + b)2 = a2 + 2ab + b2, (a − b)2 = a2 − 2ab + b2, (a + b)(a − b) = a2 − b2. Polynomial division reverses multiplication: dividend ÷ divisor = quotient, remainder R, with dividend = (divisor)(quotient) + R.

Why it matters. Special products are the fast lane of algebra — spotting them saves whole lines of FOIL. Division is how you factor higher-degree polynomials and find zeros: the Remainder Theorem says the remainder of f(x) ÷ (x − r) is simply f(r).

Where it is used. Factoring cubics by testing roots · simplifying rational expressions · finding asymptotes of rational functions.

2 See It

Diagrams that make the idea visual.

The square pattern, seen

(x + 5)2 is not x2 + 25 — the middle term 2·x·5 = 10x comes from the two cross terms of FOIL. Every perfect square has three terms: first squared, twice the product, last squared.

(x + 5)² = x² + 10x + 25 x² 2·x·5 = 10x 5² = 25 first² twice the product last² (a + b)² = a² + 2ab + b² (a − b)² = a² − 2ab + b²
(x + 5)² = x² + 10x + 25: first squared, twice the product, last squared.

Division undoes multiplication

Dividing x3 + 2x2 − 5x − 6 by x + 1 gives quotient x2 + x − 6 with remainder 0. Check: (x + 1)(x2 + x − 6) expands back to the dividend — division is multiplication in reverse.

dividend = (divisor)(quotient) + R x³ + 2x² − 5x − 6 = (x + 1)(x² + x − 6) + 0 R = 0 means x + 1 is a factor always multiply back to check a division
Dividend = (divisor)(quotient) + remainder. R = 0 means the divisor is a factor.

3 Worked Examples

Follow each step. The pattern is always the same.

Example 1 Expand: (x + 5)²
  1. Pattern: (a + b)2 = a2 + 2ab + b2.
  2. a2 = x2; 2ab = 2·x·5 = 10x; b2 = 25.
(x + 5)² = x² + 10x + 25

Check: FOIL gives the same four terms: x² + 5x + 5x + 25.

Example 2 Expand: (x − 3)²
  1. Pattern: (a − b)2 = a2 − 2ab + b2.
  2. x2 − 2·x·3 + 32 = x2 − 6x + 9.
(x − 3)² = x² − 6x + 9

Check: The middle term is negative, the last term stays positive.

Example 3 Expand: (x + 4)(x − 4)
  1. Pattern: (a + b)(a − b) = a2 − b2 — the middle terms cancel.
  2. x2 − 42 = x2 − 16.
(x + 4)(x − 4) = x² − 16

Check: FOIL: x² − 4x + 4x − 16; the middle pair cancels exactly.

Example 4 Divide: (x³ + 2x² − 5x − 6) ÷ (x + 1)
  1. x3 ÷ x = x2; multiply back and subtract. Repeat with the new leading term.
  2. Continuing gives quotient x2 + x − 6 with nothing left over.
(x³ + 2x² − 5x − 6) ÷ (x + 1) = x² + x − 6, remainder 0

Check: (x + 1)(x² + x − 6) = x³ + 2x² − 5x − 6. Matches.

Example 5 Synthetic: (2x³ − 3x² + 4x − 5) ÷ (x − 2)
  1. Synthetic with 2: bring down 2; ×2 = 4, add to −3 → 1; ×2 = 2, add to 4 → 6; ×2 = 12, add to −5 → 7.
  2. Quotient 2x2 + x + 6, remainder 7.
(2x³ − 3x² + 4x − 5) ÷ (x − 2) = 2x² + x + 6, remainder 7

Check: Remainder Theorem: f(2) = 16 − 12 + 8 − 5 = 7. Matches.

4 Common Mistakes

These errors show up on almost every quiz. Spot them now and they will never cost you points.

Mistake 1: (a + b)² = a² + b²
Wrong
Dropping the middle term: (x + 5)2 = x2 + 25.
Right
x2 + 10x + 25 — the 2ab term is the whole point of the pattern.
Rule: (a + b)² = a² + 2ab + b². Never skip 2ab.
Mistake 2: sign slips in long division
Wrong
Subtracting the partial product but forgetting to flip its signs.
Right
Each subtraction flips every sign of the partial product before adding.
Rule: “subtract” in long division means add the opposite.
Mistake 3: ignoring the remainder
Wrong
Writing quotient only when the division leaves a remainder.
Right
Report quotient and remainder: dividend = (divisor)(quotient) + R.
Rule: the remainder is part of the answer — R = 0 is the factor test.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Expand (x + 7)² using the special-product pattern.
Answer
x2 + 14x + 49.
2. Divide (x³ − 8) ÷ (x − 2).
Answer
Quotient x² + 2x + 4, remainder 0.
3. Use synthetic division: (x³ − 4x² + 6x − 4) ÷ (x − 1).
Answer
Quotient x² − 3x + 3, remainder −1.

Key Points to Remember

  • (a + b)² = a² + 2ab + b² — the middle term 2ab is never optional.
  • (a − b)² = a² − 2ab + b² — middle negative, last positive.
  • (a + b)(a − b) = a² − b² — the middle terms cancel.
  • Long division: divide leading terms, multiply back, subtract (flip all signs), repeat.
  • Synthetic division is the shortcut for divisors of the form x − r.
  • Remainder Theorem: f(x) ÷ (x − r) leaves remainder f(r).
  • R = 0 means the divisor is a factor of the dividend.
Practice problems Try the tool See it on FiboMap
Fibo · Learn, Practice & Explore Math