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Factoring A Trinomial with a Negative Sign

Examine the expression below.

6c2 + c - 12

The expression is a trinomial, like the expressions which we factored on previous pages. There is a small variation in this problem however -- the first term has a coefficient that is not 1.

This expression will be factored much like the others, the main difference is that we will need to find the correct pair of factors for the first term's coefficient (6) in addition to the correct pair of factors for the last term's coefficient (-12).

Start out by writing two sets of empty parentheses.

(            )(            )

Write each variable of the first and last terms in their respective positions, but with only half the exponent. The first term has the variable c2, with half its exponent it becomes c which is written on the left of each set of parentheses. The last term does not have any variables to write.

(  c        )(  c        )

Because the last term is negative, write a plus sign in the middle of the first set of parentheses and a - sign in the middle of the second. (If the last term were positive you would write a + sign in each.)

(  c +     )(  c -     )

Now write out the positive factors of the coefficient of first and last.

Factors of 6
1 * 6
2 * 3
Factors of -12
1 * -12
-1 * 12
2 * -6
-2 * 6
3 * -4
-3 * 4

To find the right combination of factors, we will use Trial and Error as in the last two lessons, but we must try different factors for both the first and last terms. Each pair of factors must be tried in two arrangements. That is the pair "1 * 6" must be tried as both "1 * 6" and "6 * 1". The work below will explain this a little better.

After you write out a possible combination, FOIL the factored polynomial. If the result is eqivalent to the original expression, you have found the answer. All of the possible combinations for the above problem are shown below.

(1c + 1)(6c - 12)
(1c + 12)(6c - 1)
(1c + 2)(6c - 6)
(1c + 6)(6c - 2)
(1c + 3)(6c - 4)
(1c + 4)(6c - 3)
(2c + 1)(3c - 12)
(2c + 12)(3c - 1)
(2c + 2)(3c - 6)
(2c + 6)(3c - 2)
(2c + 3)(3c - 4)
(2c + 4)(3c - 3)

After using the FOIL Method to check each possible solution above, you should find that (2c + 3)(3c - 4) is the correct factorization.

Proceed to the next page for various resources for this lesson.

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