A trinomial is an algebraic expression made up of three terms. Most likely, you'll start learning how to factor quadratic trinomials, meaning trinomials written in the form ax2 + bx + c. There are several tricks to learn that apply to different types of quadratic trinomial, but you'll get better and faster at using them with practice. Higher degree polynomials, with terms like x3 or x4, are not always solvable by the same methods, but you can often use simple factoring or substitution to turn them into problems that can be solved like any quadratic formula.
Method 1: Factoring x2 + bx + c
1. Learn FOIL multiplication. You might have already learned the FOIL method, or "First, Outside, Inside, Last," to multiply expressions like (x+2)(x+4). It's useful to know how this works before we get to factoring:
- 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
2. Understand factoring. When you multiply two binomials together in the FOIL method, you end up with a trinomial (an expression with three terms) in the form ax2+bx+c, where a, b, and c are ordinary numbers. If you start with an equation in the same form, you can factor it back into two binomials.
- 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."
3. Write a space for the answer in FOIL form. For now, just write (__ __)(__ __) in the space where you'll write the answer. We'll fill this out as we go.
- Don't write + or - between the blank terms yet, since we don't know which it will be.
4. Fill out the First terms. For simple problems, where the first term of your trinomial is just x2, the terms in the First position will always be x and x. These are the factors of the term x2, since x times x = x2.
- 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.
5. Use factoring to guess at the Last terms. If you go back and reread the FOIL method step, you'll see that multiplying the Last terms together gives you the final term in the polynomial (the one with no x). So to factor, we need to find two numbers that multiply to form the last term.
- 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 __).
6. Test which possibilities work with Outside and Inside multiplication. We've narrowed the Last terms down to a few possibilities. Use trial and error to test each possibility, multiplying the Outside and Inside terms, and comparing the result to our trinomial. For example:
- 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.