3 Ways To Factor Trinomials

Multiply the First terms: (x+2)(x+4) = x2 + __ Multiply the Outside terms: (x+2)(x+4) = x2+4x + __ Multiply the Inside terms: (x+2)(x+4) = x2+4x+2x + __ Multiply the Last terms: (x+2)(x+4) = x2+4x+2x+8 Simplify: x2+4x+2x+8 = x2+6x+8

If the equation isn’t written in this order, move the terms around so they are. For example, rewrite 3x - 10 + x2 as x2 + 3x - 10. Because the highest exponent is 2 (x2), this type of expression is “quadratic. "

Don’t write + or - between the blank terms yet, since we don’t know which it will be.

Our example x2 + 3x - 10 just begins with x2, so we can write: (x __)(x __) We’ll cover more complicated problems in the next section, including trinomials that begin with a term like 6x2 or -x2. For now, follow the example problem.

In our example x2 + 3x - 10, the last term is -10. What are the factors of -10? What two numbers multiplied together equal -10? There are a few possibilities: -1 times 10, 1 times -10, -2 times 5, or 2 times -5. Write these pairs down somewhere to remember them. Don’t change our answer yet. It still looks like this: (x __)(x __).

Our original problem has an “x” term of 3x, so that’s what we want to end up with in this test. Test -1 and 10: (x-1)(x+10). The Outside + Inside = 10x - x = 9x. Nope. Test 1 and -10: (x+1)(x-10). -10x + x = -9x. That’s not right. In fact, once you test -1 and 10, you know that 1 and -10 will just be the opposite of the answer above: -9x instead of 9x. Test -2 and 5: (x-2)(x+5). 5x - 2x = 3x. That matches the original polynomial, so this is the correct answer: (x-2)(x+5). In simple cases like this, when you don’t have a constant in front of the x2 term, you can use a shortcut: just add the two factors together and put an “x” after it (-2+5 → 3x). This won’t work for more complicated problems, though, so it’s good to remember the “long way” described above.

3x2 = (3)(x2) 9x = (3)(3x) -30 = (3)(-10) Therefore, 3x2 + 9x - 30 = (3)(x2+3x-10). We can factor out the new trinomial using the steps in the section above. Our final answer will be (3)(x-2)(x+5).

2x2y + 14xy + 24y = (2y)(x2 + 7x + 12) x4 + 11x3 - 26x2 = (x2)(x2 + 11x - 26) -x2 + 6x - 9 = (-1)(x2 - 6x + 9) Don’t forget to factor the new trinomial further, using the steps in method 1. Check your work and find similar example problems in the example problems near the bottom of this page.

Set up our answer: (__ )( __) Our “First” terms will each have an x, and will multiply together to make 3x2. There’s only one possible option here: (3x __)(x __). List factors of 8. Our options are 1 times 8, or 2 times 4. Test these using the Outside and Inside terms. Note that the order of the factors matter, since the Outside term is being multiplied by 3x instead of x. Try out every possibility until you get an Outside+Inside result of 10x (from the original problem): (3x+1)(x+8) → 24x+x = 25x no (3x+8)(x+1) → 3x+8x = 11x no (3x+2)(x+4) → 12x+2x=14x no (3x+4)(x+2) → 6x+4x=10x yes This is the correct factor.

x5+13x3+36x =(x)(x4+13x2+36) Let’s invent a new variable. We’ll say y = x2, and plug it in: (x)(y2+13y+36) =(x)(y+9)(y+4). Now switch back to using the original variable: =(x)(x2+9)(x2+4) =(x)(x±3)(x±2)

For example, in x2 + 6x + 5, 5 is a prime number, so the binomial must be in the form (__ 5)(__ 1). In the problem 3x2+10x+8, 3 is a prime number, so the binomial must be in the form (3x __)(x __). For the problem 3x2+4x+1, the only possible solution is (3x+1)(x+1). (You should still multiply this out to check your work, since some expressions can’t be factored at all – for example, 3x2+100x+1 has no factors. )

x2+2x+1=(x+1)2, and x2-2x+1=(x-1)2 x2+4x+4=(x+2)2, and x2-4x+4=(x-2)2 x2+6x+9=(x+3)2, and x2-6x+9=(x-3)2 A perfect square trinomial in the form ax2 + bx + c always has a and c terms that are positive perfect squares (such as 1, 4, 9, 16, or 25), and a b term (positive or negative) that equals 2(√a * √c). [2] X Research source

For non-quadratic trinomials, use Eisenstein’s Criterion, described in the Tips section.

(2y)(x2 + 7x + 12) = (x+3)(x+4) (x2)(x2 + 11x - 26) = (x+13)(x-2) (-1)(x2 - 6x + 9) = (x-3)(x-3) = (x-3)2

3x3+3x2-6x = (3x)(x+2)(x-1) ← highlight that space to see the answer -5x3y2+30x2y2-25y2x = (-5xy^2)(x-5)(x-1)

2x2+3x-5 = (2x+5)(x-1) ← highlight to see answer 9x2+6x+1 = (3x+1)(3x+1)=(3x+1)2 (Hint: You might need to try more than one pair of factors for 9x. )