Factoring
Grades 9–11 Algebra Polynomials
Factoring is distributing in reverse: instead of
3(x + 4) → 3x + 12, you go
3x + 12 → 3(x + 4). It breaks an expression into the
pieces that multiply to make it — essential for solving quadratics.
Level 1: pull out the common factor
6x + 18 = 6(x + 3) ← 6 divides both terms
4x² + 8x = 4x(x + 2)
Level 2: x² + bx + c
Find two numbers that multiply to c and add to b:
x² + 7x + 12
? · ? = 12 ? + ? = 7 → 3 and 4 work: 3·4=12, 3+4=7
x² + 7x + 12 = (x + 3)(x + 4)
Check: (x+3)(x+4) = x² + 4x + 3x + 12 = x² + 7x + 12 ✓
Level 3: difference of squares
x² − 9 = (x + 3)(x − 3) ← a² − b² = (a+b)(a−b)
x² − 25 = (x + 5)(x − 5)
Sign rules for x² + bx + c
+c → both factors same sign (sign of b). −c → opposite signs,
the bigger number takes b's sign. x² − 2x − 15 → factors of −15
adding to −2 → −5 and +3 → (x − 5)(x + 3).
Practice
x² + 5x + 6 factors as (x + 3)(x + ___)
x² − 2x − 15 factors as:
x² − 49 = (x + 7)(x − ___)
10x + 35 = 5(2x + ___)
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