Trinomials are polynomials with three terms. Some neat tricks are available for factoring trinomials; all of these methods involve your ability to factor a number into all its possible pairs of factors. It is worth repeating that for these problems it is crucial to remember that you must consider all possible pairs of factors and not just prime factors. For example, if you are factoring the number 24, all possible pairs are 1, 24; 2, 12; 3, 8 and 4, 6.
Pay attention to the order in which the trinomial is written. Make sure you write it in descending order, which means highest exponent of variables (such as "x") on the left going down sequentially as you move right.
Example 1: – 10 - 3x+ x^2 must be rewritten as x^2 - 3x – 10
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Example 2: – 11x + 2x^2 – 6 must be rewritten as 2x^2 – 11x – 6
Remember to take out all factors common to all terms in the trinomial. The common factor is called the GCF (Greatest Common Factor).
Example 1: 2x^3y – 8x^2y^2 – 6xy^3 \= (2xy)x^2 – (2xy)4xy – (2xy)3y^2 \= 2xy(x^2 – 4xy - 3y^2)
Try to factor further if possible. In this case, the remaining trinomial cannot be factored further; hence that is the answer in its most simplified form.
Example 2: 3x^2 – 9x – 30 \= 3(x^2 - 3x – 10) You can factor this trinomial (x^2 - 3x – 10) further. The correct answer to the problem is 3(x + 2)(x – 5); the method for achieving this is discussed in Section 3.
Trick 1 - Trial and Error
Consider the trinomial (x^2 - 3x – 10). Your goal is to break up the number 10 into pairs of factors in such a way that when you add those two factors of 10, they have a difference of 3, which is the coefficient of the middle term. In order to get this, you know that one of the two factors will be positive, the other negative. Clearly write (x + )( x - ) leaving a space for the second term in each parentheses. The pairs of factors of 10 are 1, 10 and also 2, 5. The only way to get -3 by adding the two factors is to choose -5 and 2. This way you get -3 for the coefficient of the middle term. Fill in the empty spots. Your answer is (x + 2)(x – 5)
Trick 2 – British Method
This method is helpful when the trinomial has a leading coefficient, such as 2x^2 – 11x – 6, where 2 is the "leading" coefficient because it belongs to the leading, or first, variable. The leading variable is the one with the highest exponent and must always be written first and sit on the left.
Multiply the first term (2x^2) and the last term (6), without their signs, to get the product 12x^2. Factor the coefficient 12 into all possible pairs of factors, regardless of whether they are prime. Always start with 1. Your factors should be 1, 12; 2, 6 and 3, 4. Take each pair and see whether it yields the coefficient of the middle term -11, when you add or subtract them. When you select 1 and 12, a subtraction yields 11. Adjust the sign accordingly; in this problem the middle term is -11x, therefore the pairs must be -12x and 1x, which is simply written as x.
Write all terms clearly: 2x^2 – 12x + x – 6 For each pair of terms, factor out common terms. 2x(x – 6) + (x – 6) or 2x(x – 6) + (1)(x – 6)
Factor out common factors. (x – 6)(2x + 1)
After you have completed the factoring, use FOIL (the first, inner, outer, last method of multiplying two binomials) to check whether you have the correct answer. You should get the original polynomial when you use FOIL to confirm your factoring is correct.