Learn · Algebra 1

Polynomials & Factoring: FOIL, GCF, and the Three Patterns

Multiply binomials with FOIL, then run the process backward to factor. Three patterns cover most CBE factoring questions: GCF, trinomial (x² + bx + c), and difference of squares.

9 min TEKS 10A,10B,10D,10E,10F Algebra 1

Multiplying and factoring are mirror operations

Multiplying binomials with FOIL turns (x + 2)(x + 3) into x² + 5x + 6. Factoring runs the same operation backward — turning x² + 5x + 6 back into (x + 2)(x + 3). One direction is mechanical; the other is the most-tested skill in Algebra 1.

FOIL: First, Outer, Inner, Last

(x + 2)(x + 3) F: x · x = x² L: 2 · 3 = 6 I: 2 · x = 2x O: x · 3 = 3x = x² + 5x + 6
F → First terms. O → Outer. I → Inner. L → Last. Add all four products.

Pattern 1: Always pull the GCF first

Before any other factoring move, ask: “Do all terms share a factor?” If yes, factor it out.

3x² − 12   →   GCF is 3 = 3(x² − 4) Now factor x² − 4 (difference of squares!) = 3(x − 2)(x + 2)
Factor with GCF first
Factor: 3x²−12

Pattern 2: Trinomial (x² + bx + c)

For x² + bx + c, find two numbers that multiply to c and add to b.

x² + 8x + 15 Need: two numbers, product 15, sum 8 3 · 5 = 15, 3 + 5 = 8 ✓ = (x + 3)(x + 5)
Sign-reading shortcut

c > 0 and b > 0: both factors positive. c > 0 and b < 0: both factors negative. c < 0: one positive, one negative — the bigger absolute value gets the sign of b.

Factor a positive-c trinomial
Factor: x²+8x+15
Factor a negative-b trinomial
Factor: x² − 7x + 12

Trinomial when leading coefficient ≠ 1

For ax² + bx + c when a ≠ 1, use the AC method: find two numbers that multiply to a·c and add to b, then split the middle term.

2x² + 7x + 3 a·c = 6,   need product 6, sum 7 → 6 and 1 2x² + 6x + x + 3 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3)
Factor with leading coefficient
Factor: 2x² + 7x + 3

Pattern 3: Difference of squares

a² − b² = (a − b)(a + b) Two perfect squares with a minus sign between them. Sum of squares does NOT factor over real numbers.
Spotting the pattern

Look for two terms, both perfect squares, with a minus sign between. Examples: x² − 9 = (x − 3)(x + 3). 4y² − 25 = (2y − 5)(2y + 5).

Word-problem application

Factor a polynomial in context
The area of a rectangular yard is given by the polynomial 2x² + 7x + 3 square feet. Which pair of binomials could represent the length and width of the yard?
Area = 2x² + 7x + 3 square feet length = ? width = ? Find binomial factors of the area expression.

3-second recap

  • FOIL → First, Outer, Inner, Last (multiply binomials)
  • Always pull the GCF first before any other factoring
  • Trinomial x² + bx + c → find two numbers with product c, sum b
  • Leading coefficient ne 1 → AC method (split the middle term)
  • Difference of squares: a² − b² = (a − b)(a + b)

Check yourself

Quick check #1
Factor: x² − 9.
Quick check #2
Multiply: (x + 5)(x − 2).