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Daily Math Minute

Factoring Polynomials

Factoring Trinomials

Factoring quadratic trinomials of the form x² + bx + c and ax² + bx + c.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

  • Factoring the GCF

Reversing FOIL

Recall from multiplying polynomials that (x + 3)(x + 5) expands to x² + 8x + 15 — the 8 came from adding 3 and 5, and the 15 came from multiplying them. Before reading on, think about this in reverse: given x² + 7x + 12, what two numbers would need to both add to 7 and multiply to 12?

Testing pairs that multiply to 12 — 1 and 12, 2 and 6, 3 and 4 — only 3 and 4 also add to 7. That means x² + 7x + 12 factors as (x + 3)(x + 4), and checking with FOIL confirms it: x² + 4x + 3x + 12 = x² + 7x + 12.

Worked Example — Factoring with a Negative Middle Term

Factor x² − 8x + 15. Find two numbers that multiply to 15 and add to −8. Since the product is positive and the sum is negative, both numbers must be negative: −3 and −5 multiply to 15 and add to −8. The factored form is (x − 3)(x − 5).

Worked Example — Factoring with a Negative Constant Term

Factor x² + 2x − 15. Find two numbers that multiply to −15 and add to 2. Since the product is negative, the numbers have opposite signs: 5 and −3 multiply to −15 and add to 2. The factored form is (x + 5)(x − 3).

When the leading coefficient isn't 1, like in 2x² + 7x + 3, the same number pair idea still applies, just adjusted: find two numbers that multiply to (leading coefficient × constant term) and add to the middle coefficient, then use those numbers to split the middle term and factor by grouping.

Worked Example — Factoring with a Leading Coefficient Other Than 1

Factor 2x² + 7x + 3. Multiply the leading coefficient and constant: 2 × 3 = 6. Find two numbers that multiply to 6 and add to 7: 1 and 6. Split the middle term: 2x² + x + 6x + 3. Group and factor each pair: x(2x + 1) + 3(2x + 1). Factor out the shared (2x + 1): (2x + 1)(x + 3).

Tip

For a trinomial with leading coefficient 1, the two numbers you're searching for become the constants inside the two binomial factors directly — no extra grouping step needed.

Common Mistakes

  • Finding two numbers that add correctly but multiply incorrectly, or vice versa, without checking both conditions together.

    Both conditions — the correct sum and the correct product — must hold at the same time. Test candidate pairs against both requirements before finalizing.

  • Forgetting to check the sign pattern first (positive/negative constant term) before searching for the number pair.

    A positive constant term means the two numbers share the same sign (matching the middle term's sign); a negative constant term means they have opposite signs — checking this first narrows the search significantly.

Key Takeaways

  • Factoring a trinomial x² + bx + c means finding two numbers that multiply to c and add to b.
  • The signs of b and c determine whether the two numbers share a sign or have opposite signs.
  • When the leading coefficient isn't 1, the same number-pair search (scaled by the leading coefficient) sets up factoring by grouping.

Summary

Factoring trinomials reverses the FOIL pattern by searching for a matching sum-and-product pair. The final lesson in this unit recognizes two special trinomial patterns that factor especially quickly.

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