Factoring Trinomials Worksheet – When a is Not 1

Factor a trinomial whose leading coefficient is greater than 1, such as 3x² + 10x − 8 = (3x − 2)(x + 4)

Factor trinomials where both binomial factors carry a coefficient, as with 20x² − 96x + 27 = (10x − 3)(2x − 9)

Recognise a perfect square trinomial, as in 9x² + 24x + 16 = (3x + 4)(3x + 4)

Use a matching bank as a self-check, treating a letter that is already taken as a signal to revisit the working

Handle negative constant and middle terms confidently across signs, from 2x² − 7x − 30 to 7x² − 31x + 30

The two pages are graded deliberately: on page one only one factor of each pair is monic, so the bank gives away the easier half, while page two's bank lists binomials that both carry coefficients

Because each letter is used exactly once, a student who finds a letter already taken knows immediately that one of the two answers is wrong — worth pointing out before they start, since it turns the sheet into its own feedback loop

Problem 18, 9x² + 24x + 16, factors as (3x + 4)(3x + 4), which makes it the natural example for spotting a perfect square trinomial before grinding through the general method

Problem 13, 20x² − 96x + 27, has the largest coefficients on the sheet and the most factor pairs to test, so it is the one worth modelling or leaving as an extension

The key prints both factors rather than the letter alone, so it can be used to mark the working as well as the matching

Factoring trinomials with a leading coefficient above 1 — every one of the eighteen problems has an x² coefficient between 2 and 27, so the simple sum-and-product method will not work.

Testing factor pairs systematically — problems such as 20x² − 96x + 27 and 27x² + 66x − 16 require working through the factor pairs of both the leading coefficient and the constant.

Managing signs in a factorisation — the set mixes all four sign patterns, from 5x² + 19x + 18 through 6x² − 29x − 5 to 7x² − 31x + 30.

Recognising a perfect square trinomial — 9x² + 24x + 16 resolves to two identical factors, the only such case on the sheet.

Self-checking against a fixed bank — each lettered binomial is used exactly once across nine problems, so a clash reveals an error without a marker being involved.