L5 · Factoring x^2 + bx + c

Learning goal

“I can factor x^2 + bx + c by finding the pair of integers that multiplies to c and adds to b, and verify every factorization by expanding.”

Factor pair — two integers whose product is a given number (3 and 4 are a factor pair of 12).

Constant term — the term with no variable (the c in x^2 + bx + c); it equals the product of the two binomials' constants.

Middle coefficient — the b in x^2 + bx + c; it equals the SUM of the two binomials' constants.

Factored form — (review) the product form: (x + p)(x + q), where p * q = c and p + q = b.

Worked example

Factor x^2 + 7x + 12. Step 1: The answer will look like (x + p)(x + q) (tiles: a rectangle with x-tiles along both sides). Step 2: From L2, p * q = 12 (constant) and p + q = 7 (middle). Step 3: Pairs of 12: (1, 12) sums 13; (2, 6) sums 8; (3, 4) sums 7. Step 4: x^2 + 7x + 12 = (x + 3)(x + 4). Step 5: Verify by expanding: x^2 + 4x + 3x + 12 = x^2 + 7x + 12. Sign rule for c > 0: p and q have the SAME sign - both positive if b > 0, both negative if b < 0.

Check yourself — explain it

For x^2 - 10x + 21, both constants in the factors must be negative because...

Paired problem — example still visible

Factor x^2 + 9x + 20 -> (x + 4)(x + 5) (4 * 5 = 20; 4 + 5 = 9)

Faded completion — last step is yours

"x^2 - 8x + 15 = (x - ___)(x - 3)" -> 5 (the pair is (-5, -3): product +15, sum -8)

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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Grade 10 — Mathematics 10C › C10U06 Polynomials: Multiply & Factor

L5 · Factoring x^2 + bx + c