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Factoring A Trinomial Lessons

This lesson explains how to factor trinomials. The process presented is essentially the opposite of the FOIL Method, which is a process used to multiply two binomials. Make sure you understand the FOIL Method lesson first.

Examine the following expression which consists of one binomial in parentheses multiplying another binomial in parentheses.

(2k + 7)(3k - 10)

Using the FOIL Method, we can simplify the expression, obtaining the work below and a final result which is a trinomial.

6k2 - 20k + 21k - 70
6k2 + k - 70

After the two sets of parentheses were multiplied, like terms were combined. Notice that a trinomial, which consists of three terms, remains. This lesson explains how to factor a trinomial like this into two binomials (which we had before the FOIL Method was applied).

The first problem we will examine is below.

-16 - m2 -10m

In this problem, all of the terms are negative. Therefore, our first step is to factor out the Greatest Common Factor which is a -1 or simply a minus sign.

-(16 + m2 + 10m)

To make the factoring process a little more consistent and easier, it is a good idea to keep the terms in order by the variable's exponent. Since m is the only variable letter in this expression we will order the terms from the highest power of m to the lowest power of m.

-(m2 + 10m + 16)

Before attempting to factor any more, there are a few simple questions you can ask to make sure that the expression is factorable as a trinomial.

Question Answer and Reason
After the GCF has already been factored out, are all variables in the first (m2) and last (16) terms to an even power? Yes. (m2 is the only variable in the first and last terms, and its exponent, 2, is an even number.)
Are all of the middle term's (10m) exponents to half of the power of one of the exponents on the outside terms? Yes. (In 10m, the middle term, m is the only variable. Its exponent is not shown and therefore is 1 which is half of 2, the exponent of m in the first term.)

The next step is to carry down the minus sign and write sets of open parentheses, side by side.

-(          )(          )

Handling Variables

On the left side of each set of parentheses, write all of the variables from the first term of the trinomial with half their exponents. The first term is m2, thus after the exponent is cut in half, m is placed inside each set of parentheses.

-(  m      )(  m      )

Since all of the terms in the original set of parentheses is positive, two plus signs are placed below.

-(  m +    )(  m +    )

Handling Coefficients

Now identify the coefficient of the first and last terms in the original set of parentheses, m2 + 10m + 16. The coefficients are 1 and 16. Now write all pairs of factors of 1 in a vertical column and then write all pairs of factors of 16 in another vertical column.

Factors of 1
1 * 1
Factors of 16
1 * 16
2 * 8
4 * 4

Choosing Factor Pairs

Now we must choose a pair of factors of 1 and a pair of factors of 16 to insert into the pairs of parentheses. But how is this done? For the number 1, there is only 1 pair of factors, eliminating any choice. Thus, the 1s can be inserted on the left side of each set of parentheses.

-(1m +    )(1m +    )

Finally, a pair of factors of 16 must be chosen. This is a matter of trial and error. To make sure all pairs are considered, start using factors at the top of the list then check each pair below it, in order.

We start by choosing the first pair, 1 * 16. We place the factors on the right side of each set of parentheses.

-(1m +  1 )(1m +  16)

Now test whether this is the correct pair. Multiply the expression using the FOIL Method:

-(1m2 + 16m + m + 16)
-(1m2 + 17m + 16)

Note that the resulting trinomial is not the same as the one we started with, so this is the incorrect pair of factors of 16. Try the next pair:

-(1m +  2 )(1m +  8 )
-(1m2 + 8m + 2m + 16)
-(1m2 + 10m + 16)

Since the result here is eqivalent to the original, the correct pair of factors of 16 has been chosen. Additionally, the answer to the problem is.

-(1m + 2)(1m + 8)

Proceed to the next page to examine another example.


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