Factoring a Trinomial as the Product of Two Binomials
Factoring x2+bx+c means finding two numbers that multiply to c and add to b; those numbers become the constants in (x+p)(x+q).
When you multiply (x+p)(x+q), you get x2 + (p+q)x + pq. The middle term is the sum of p and q, and the last term is their product. So factoring a trinomial just reverses that: you already know the sum and product you need, and you hunt for the two numbers that give them. For x2+7x+12, you want two numbers adding to 7 and multiplying to 12 — that's 3 and 4, so the trinomial is (x+3)(x+4).
Picture a rectangle with area x2+7x+12, built from an x-by-x square, two strips of width x, and a row of unit squares. Splitting it into a grid with side lengths (x+3) and (x+4) shows exactly how the pieces fit — the strips split into 3x and 4x, and the small squares total 12. That's the same logic that makes grouping work on harder trinomials like 6x2+11x-10: split 11x into two pieces (15x and -4x, since 15 times -4 is -60, matching 6 times -10, and 15-4=11), then group (6x2+15x) + (-4x-10), pull out common factors to get 3x(2x+5) - 2(2x+5), and factor out (2x+5) to land on (2x+5)(3x-2).
This connects directly to solving quadratic equations: once a trinomial is factored, setting each binomial to zero gives the roots. For x2-2x-35, the numbers -7 and 5 multiply to -35 and add to -2, so it factors as (x-7)(x+5), and the equation x2-2x-35=0 has solutions x=7 and x=-5. Not every trinomial cooperates, though — that's where you either accept it doesn't factor over integers or move on to the quadratic formula.
Key terms
- Trinomial
- A polynomial with three terms, such as x2+7x+12.
- Binomial
- A polynomial with two terms, such as x+3.
- Grouping
- A factoring method that splits a four-term expression into two pairs, each with a common factor.
- Factor over the integers
- Breaking an expression into pieces whose coefficients are all whole numbers.
Watch it explained
- How it worksHow do the sum and the product of two hidden numbers surface as the middle and last terms of a trinomial?
- How it worksWalk me through the area-rectangle picture that turns x2+7x+12 into (x+3)(x+4).
- Worked exampleFind the factorization of x2-2x-35 and give both solutions of x2-2x-35=0.
- Worked exampleShow me how to factor 6x2+11x-10 into two binomials using the grouping method.
- Common mistakeWhy doesn't every trinomial with whole-number coefficients break into binomials with whole-number coefficients?
Check yourself
End of topic.