Algebraic Expressions • Topic 3 of 4

Common Factoring Patterns

Factoring runs expansion in reverse, and the GRE recycles only a handful of patterns. Always pull out the greatest common factor first: 6x² + 9x = 3x(2x + 3). Recognise a difference of squares instantly — a² − b² = (a + b)(a − b), so x² − 16 = (x + 4)(x − 4) and even 9x² − 25 = (3x + 5)(3x − 5). For a simple trinomial x² + bx + c, find two numbers that multiply to c and add to b; for x² + 7x + 12 that pair is 3 and 4, giving (x + 3)(x + 4). Perfect-square trinomials collapse to (x ± k)². Factoring is the fastest route through Quantitative Comparison and many "which must be true" questions, because a factored form exposes roots and shared factors that a messy polynomial hides.

✅ Solved examples

1. Factor x² − 49.
Difference of squares: x² − 7² = (x + 7)(x − 7).
2. Factor x² + 9x + 20.
Need two numbers multiplying to 20 and adding to 9: 4 and 5. So (x + 4)(x + 5).
3. Factor 12x² − 8x completely.
GCF is 4x: 12x² − 8x = 4x(3x − 2).
4. Factor x² − 10x + 25.
A perfect-square trinomial: 25 = 5², and 2·5 = 10, so it is (x − 5)².

✏️ Practice — try these, take hints as needed

1. Factor x² − 36.
Difference of squares.
a = x, b = 6.
(x + 6)(x − 6).
(x + 6)(x − 6)
2. Factor x² + 8x + 15.
Product 15, sum 8.
Try 3 and 5.
(x + 3)(x + 5).
(x + 3)(x + 5)
3. Factor x² − 2x − 15.
Product −15, sum −2.
Try +3 and −5.
Signs differ.
(x + 3)(x − 5)
4. Factor 5x² + 15x completely.
Find the GCF.
5x is common.
5x(x + 3).
5x(x + 3)
5. Factor 9x² − 1.
Difference of squares.
9x² = (3x)², 1 = 1².
(3x + 1)(3x − 1).
(3x + 1)(3x − 1)

📝 Topic test — 8 questions

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