How to Factor Trinomials & Binomials
You will find many skills from algebra class that you use over and over, and factoring binomials and trinomials is one of them. These problems can look quite challenging at first, but if you learn the basic steps and take each one a step at a time, you can quickly master this basic skill. Other skills in algebra build on factoring, so take the time to learn it well. The key is to look at each problem as a series of small steps, rather than one big operation.
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Instructions
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Factor binomials by factoring out a greatest common factor. For instance, if you are factoring 2x^2 - 50, you can factor out a 2. This gives 2(x^2 - 25).
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Check all binomials to see if they are the difference of two perfect squares. Since x^2 and 25 are both perfect squares, 2 (x^2 - 25) can be factored further.
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Factor a problem containing the difference of squares by plugging it into this pattern: ( ___ x - ___ ) ( ___ x + ___ ). The numbers in the blanks are the square roots of the numbers in the binomial. The square root of x^2 is x, and the square root of 25 is 5, so this will look like 2 (x - 5) (x +5).
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Factor trinomials that do not have a leading coefficient by first listing all of the factors of the last term. If you are factoring x^2 + 3x - 40, the factors of -40 are 1 * -40, -1 * 40, 2 * -20, -2 * 20, 4 * -10, -4 * 10, 5 * -8, and -5 * 8.
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Determine which pair of factors will add together to give you the middle term, and fit those numbers into the basic factoring pattern (x +/- ____ ) (x +/- ____ ). In this example, the pair you will use is -5 * 8, which looks like (x - 5) (x + 8).
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Factor trinomials that do have a leading coefficient by first factoring out a greatest common factor if there is one. In 12x^2 - 18x - 20, there is a greatest common factor of 2. Factor this out to give you 2 (6x^2 - 9x - 10).
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Factor the new trinomial by trial and error. Write out all possible factors of the first and last terms. The factors of 6 are 1 * 6 and 2 * 3. The factors of -10 are 1 * -10, -1 * 10, -2 * 5, and 5 * -2.
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Plug these possible combinations into the basic factoring pattern. You could try (2x + 1) (3x - 10).
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Use the FOIL method to check whether you have the right combination. The FOIL method asks you to multiply the First (2x and 3x), Outer (2x and -10), Inner (1 and 3x) and Last (1 and -10) terms together. This gives you 6x^2 - 20x + 3x - 10, which simplifies to 6x^2 - 17x - 10, and this is not the original trinomial.
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Keep trying combinations until you find the one that gives you the original trinomial (6x^2 - 9x - 10). The correct combination is (2x - 5) (3x + 2). Write this with the greatest common factor (2) at the front, which looks like 2 (2x - 5) (3x + 2).
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Tips & Warnings
If you factor a trinomial and get the same binomial for both of them, such as (x + 1) (x + 1), simplify this to read (x + 1)^2.
Unless a binomial is the difference of squares, you cannot factor it beyond factoring out a greatest common factor.
Use your calculator to help when you are finding the possible factors of larger numbers.
Always check your answer for the greatest common factor that you factored out at the beginning of the problem. This is a necessary part of your answer, but is easy to forget about as you work the problem.
Resources
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Comments
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oh........
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going to be good.
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go go!
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going to be good!
