Special products and factoring identities

Patterns that save time

Some products come up so often that it’s worth memorizing them. These special products work in both directions: expanding and factoring.

The three key patterns

Expanding with patterns

• (x + 9)² = x² + 18x + 81
• (4y − 3)² = 16y² − 24y + 9
• (2m + 5n)² = 4m² + 20mn + 25n²
• (x + 11)(x − 11) = x² − 121

(a + b) and (a − b) are called conjugates; multiplying them makes the middle terms cancel.

Factoring with patterns

Recognizing a perfect square trinomial:

1. Are the first and last terms perfect squares?
2. Is the middle term twice the product of their square roots?

Example: 9x² + 30x + 25

• 9x² = (3x)², 25 = 5²
• 2 × 3x × 5 = 30x ✓
• → (3x + 5)²

Recognizing a difference of squares: two perfect squares with a minus between them.

• 49a² − 64 = (7a + 8)(7a − 8)
• x⁴ − 16 = (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2)

Cubes

Memory aid SOAP: the signs in the factored form are Same, Opposite, Always Positive.

• x³ + 8 = (x + 2)(x² − 2x + 4)
• 27y³ − 1 = (3y − 1)(9y² + 3y + 1)

Mental arithmetic

• 102 × 98 = (100 + 2)(100 − 2) = 10,000 − 4 = 9,996
• 45² = (40 + 5)² = 1,600 + 400 + 25 = 2,025
• 51² − 49² = (51 + 49)(51 − 49) = 100 × 2 = 200

Worked examples

Example 1: Expand (5x − 2)² → 25x² − 20x + 4.

Example 2: Factor x² − 14x + 49 → (x − 7)².

Example 3: Factor 100 − 9p² → (10 + 3p)(10 − 3p).

Common mistakes

• (a − b)² = a² − b² (missing −2ab).
• Thinking a² + b² factors as a difference of squares does.
• Wrong signs in the cube formulas.

Try it yourself

1. Expand (x + 8)².
2. Expand (3a − 4b)(3a + 4b).
3. Factor 4x² + 12x + 9.
4. Factor 81 − x².
5. Factor x³ − 125.

Answers: 1) x² + 16x + 64 2) 9a² − 16b² 3) (2x + 3)² 4) (9 + x)(9 − x) 5) (x − 5)(x² + 5x + 25)

Quick recap

• Perfect square trinomial: (a ± b)² = a² ± 2ab + b².
• Difference of squares: a² − b² = (a + b)(a − b).
• Sum of cubes and difference of cubes follow the SOAP sign pattern.