Factoring Trinomial when Lead Coefficient is not 1

Here’s a simplified approach to factoring a trinomial when the lead coefficient (the coefficient of the x^2 term) is not 1:

Let’s factor the following.

factoring trinomial a not 1

Find Two Numbers That Multiply to the Product of the Lead Coefficient and the Last Term:

Identify the product of the lead coefficient and the last term: (a * c).

Find two numbers that add up to the coefficient of the middle term (b) and also multiply to (a * c).

In this example when you multiply 5 x -6 = -30 and the factors of -30 must add to +13.

factoring trinomial

Rewrite the Middle Term Using the Two Numbers Found in Step 1:

Replace the middle term (bx) with the two numbers found in Step 1, written as a sum.

Rewriting the trinomial.

5x^2 -2x + 15x -6

factoring trinomial

Factor by Grouping:

Group the first two terms and the last two terms.

factor trinomial
Factor each group separately, using the GCF method.

factoring trinomial

Extract Common Factors, if Applicable:

If there is a common factor (a binomial or monomial) in both groups, factor it out.
Simplify the Factored Expression:
Combine like terms and simplify the expression fully.

In this example it becomes, (x+3)(5x-2)

factor trinomial

Check Your Answer:
Multiply the factored expression to ensure it matches the original trinomial.