How To Solve Trinomials

A trinomial expression is any polynomial expression that has exactly three terms. In most cases, "solving" means factoring the expression out into its simplest components. Usually, your trinomial will either be a quadratic equation, or a higher-order equation that can be turned into a quadratic equation by factoring out variables common to all terms. Start by learning how to factor quadratics, then learn how to tackle other sorts of trinomials.

Step 1

Factor out any factors common to all terms. The equation 4x^2 + 8x + 4 has 4 as a common factor, since every term can be divided by 4. Therefore, it can be factored as 4(x^2 + 2x +1). The equation x^3 +2x^2 + x has x as a common factor. It can be factored as x(x^2 +2x +1).

Step 2

Look for any other common factors you may have missed. Sometimes, an equation has both a number and a variable that can be factored out. For example, 8x^3 +12x^2 + 16x has both 4 and x as a factor. Factored out, it becomes 4x(2x^2 + 3x + 4)

Step 3

Determine what sort of trinomial equation you have left. If the highest power of the unfactored part is a squared variable like y^2 or 4a^2, you can factor it like a quadratic equation. If your highest power term is a cubed number or higher, you have a higher order equation. By this point, you will probably not have anything greater than a cubed variable to deal with.

Step 4

Factor out the quadratic part of the equation. Many trinomial quadratics are simple sums of squares. Using an example from step one:

4x^2 + 8x + 4 = 4(x^2 + 2x + 1) = 4(x + 1)(x + 1) 4(x + 1)^2

Step 5

If you are dealing with a higher-order equation, look for a pattern that allows you to solve it like a quadratic. For example, although 4x^4 + 12x^2 + 9 looks like a tough equation at first, the answer is actually very simple: 4x^4 + 12x^2 + 9 = (2x^2 + 3)^2