Factoring Trinomials

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Skill: factoring-trinomials

Factoring Trinomials

Un-multiply polynomials: turn x² + 5x + 6 back into (x + 2)(x + 3). It is FOIL in reverse — hunt for two numbers that multiply to c and add to b, and the factors reveal themselves.

1 Understand

The core idea in plain language.

What it is. Factoring a trinomial like x2 + 5x + 6 means un-multiplying it — rewriting it as a product of two binomials: (x + 2)(x + 3). It is FOIL in reverse. For x2 + bx + c, you hunt for two numbers that multiply to c and add to b. When the x2 has a coefficient, the hunt gets slightly fancier — same idea, more candidates.

Why it matters. Factoring is the key that unlocks quadratic equations: once x2 + 5x + 6 = (x + 2)(x + 3), the solutions x = −2 and x = −3 fall out via the Zero Product Property. It is also the foundation of simplifying rational expressions and much of Algebra 2.

Where it is used. Solving projectile-motion equations · optimizing area problems · simplifying fractions with polynomials in calculus.

2 See It

Diagrams that make the idea visual.

Factoring is finding the side lengths

Factoring is finding the rectangle’s side lengths from its area pieces. The 5x splits into 2x + 3x — and 2 + 3 = 5, 2 × 3 = 6. Both conditions must hold.

x 2 x 3 x² 2x 3x 6 sides (x + 2) and (x + 3)
Factoring is finding the rectangle’s side lengths from its area pieces. The 5x splits into 2x + 3x — and 2 + 3 = 5, 2 × 3 = 6. Both conditions must hold.

The systematic hunt

List factor pairs of c, check which sums to b. Only (3, 4) gives 7 — so x2 + 7x + 12 = (x + 3)(x + 4).

pairsumverdict (1, 12)13✗ (2, 6)8✗ (3, 4)7✓ x² + 7x + 12 = (x + 3)(x + 4)
The systematic hunt: list factor pairs of c, check which sums to b. Only (3, 4) gives 7 — so x² + 7x + 12 = (x + 3)(x + 4).

3 Worked Examples

Follow each step. The pattern is always the same.

Example 1 Basic: factor x² + 5x + 6
  1. Need two numbers with product 6 and sum 5. Factor pairs of 6: (1, 6) sums to 7 ✗; (2, 3) sums to 5 ✓.
  2. x2 + 5x + 6 = (x + 2)(x + 3).
x² + 5x + 6 = (x + 2)(x + 3)

Check: FOIL back: x² + 3x + 2x + 6 = x² + 5x + 6.

Example 2 Both negative: factor x² − 7x + 12
  1. Product +12, sum −7 — both numbers must be negative (negative × negative = positive; they must add to a negative).
  2. Pairs of 12: (−3, −4) sums to −7 ✓. x2 − 7x + 12 = (x − 3)(x − 4).
x² − 7x + 12 = (x − 3)(x − 4)

Check: x² − 4x − 3x + 12 = x² − 7x + 12.

Example 3 Mixed signs: factor x² + x − 12
  1. Product −12 (numbers have opposite signs), sum +1. Pairs: (4, −3): product −12 ✓, sum +1 ✓.
  2. The bigger number takes the sign of the sum. x2 + x − 12 = (x + 4)(x − 3).
x² + x − 12 = (x + 4)(x − 3)

Check: x² − 3x + 4x − 12 = x² + x − 12.

Example 4 Leading coefficient: factor 2x² + 7x + 3
  1. Try (2x + __)(x + __): the constants multiply to 3 (1 × 3) and the cross-terms 2x·3 + 1·x must total 7x.
  2. Try 1 and 3: 6x + x = 7x ✓. 2x2 + 7x + 3 = (2x + 1)(x + 3).
2x² + 7x + 3 = (2x + 1)(x + 3)

Check: 2x² + 6x + x + 3 = 2x² + 7x + 3.

Example 5 Special case: factor x² − 9 (difference of squares)
  1. No x-term (b = 0): need numbers with product −9 and sum 0 — that is 3 and −3.
  2. x2 − 9 = (x + 3)(x − 3). Pattern: a2 − b2 = (a + b)(a − b) — memorize it.
x² − 9 = (x + 3)(x − 3)

Check: x² − 3x + 3x − 9 = x² − 9.

4 Common Mistakes

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

Mistake 1: Getting the signs wrong
Wrong
x2 − 7x + 12 = (x + 3)(x + 4). FOIL gives x2 + 7x + 12 — the middle term flipped!
Right
(x − 3)(x − 4).
Fix: If c is positive and b is negative, BOTH numbers are negative. Decide the signs before hunting for the pair.
Mistake 2: Forgetting to factor out the GCF first
Wrong
2x2 + 10x + 12 = (2x + 4)(x + 3) — technically true, but messy and usually marked wrong.
Right
Factor 2 out first: 2(x2 + 5x + 6) = 2(x + 2)(x + 3).
Memory hook: GCF first, always. It is the cheapest step and it simplifies everything after.
Mistake 3: Never checking by FOIL
Wrong
x2 + x − 12 = (x + 6)(x − 2), left unverified. (FOIL: x2 + 4x − 12 ✗ — the middle term is wrong!)
Right
(x + 4)(x − 3); FOIL confirms x2 + x − 12.
Fix: FOIL-back takes fifteen seconds and catches every bad hunt. Make it non-negotiable.

5 Quick Check

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

1. Factor x² + 8x + 15.
Answer
x2 + 8x + 15 = (x + 3)(x + 5)
Pairs of 15: (1, 15) → 16 ✗; (3, 5) → 8 ✓. FOIL-check: x² + 5x + 3x + 15. ✓
2. Factor x² − 10x + 24.
Answer
x2 − 10x + 24 = (x − 4)(x − 6)
c positive, b negative → both numbers negative. (−4) + (−6) = −10, (−4)(−6) = 24. ✓
3. Factor 3x² + 11x + 6.
Answer
3x2 + 11x + 6 = (3x + 2)(x + 3)
Cross-terms: 9x + 2x = 11x. ✓

Key Points to Remember

  • Factoring is FOIL in reverse: turn a sum back into a product.
  • For x2 + bx + c, hunt two numbers with product c and sum b.
  • GCF first, always — factor out the greatest common factor before hunting.
  • Sign rules: c > 0, b < 0 → both numbers negative; c < 0 → opposite signs, bigger takes the sum’s sign.
  • Difference of squares: a2 − b2 = (a + b)(a − b).
  • FOIL-check every answer — fifteen seconds that catches every bad hunt.
Practice problems Try the tool See it on FiboMap
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