Polynomials • Topic 4 of 4

Factoring Techniques

Factoring reverses multiplication, writing a polynomial as a product. Always check for a greatest common factor first: 6x² + 9x = 3x(2x + 3). For a quadratic x² + bx + c, find two numbers that multiply to c and add to b. Recognise the difference of squares a² − b² = (a + b)(a − b) and the perfect-square trinomial a² ± 2ab + b² = (a ± b)². Factoring is the key to solving quadratics by setting each factor to zero, simplifying rational expressions and spotting structure — one of the most valuable skills across the SAT Advanced Math domain.

✅ Solved examples

1. Factor 6x² + 9x.
Take out the GCF 3x: 3x(2x + 3).
2. Factor x² + 7x + 12.
Two numbers multiplying to 12 and adding to 7 are 3 and 4: (x + 3)(x + 4).
3. Factor x² − 16.
Difference of squares: (x + 4)(x − 4).
4. Factor x² − 6x + 9.
Perfect-square trinomial: (x − 3)².

✏️ Practice — try these, take hints as needed

1. Factor 8x² + 12x.
Find the GCF of the terms.
GCF is 4x.
Divide each term.
4x(2x + 3).
2. Factor x² + 5x + 6.
Two numbers multiply to 6, add to 5.
2 and 3.
—
(x + 2)(x + 3).
3. Factor x² − 25.
Difference of squares.
a = x, b = 5.
—
(x + 5)(x − 5).
4. Factor x² + 10x + 25.
Is it a perfect square?
25 = 5² and 2·5 = 10.
—
(x + 5)².
5. Factor x² − x − 6.
Two numbers multiply to −6, add to −1.
−3 and 2.
—
(x − 3)(x + 2).

📝 Topic test — 8 questions

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