document.write( "Question 893955: factor the trinomial \r
\n" ); document.write( "\n" ); document.write( " $\"c%5E2%2B6c-16\"$ \n" ); document.write( "

Algebra.Com's Answer #541652 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
already answered today
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"c%5E2-6c-16\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-6\"$ , and the last term is $\"-16\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-16\"$ to get $\"%281%29%28-16%29=-16\"$ .

Now the question is: what two whole numbers multiply to $\"-16\"$ (the previous product) and add to the second coefficient $\"-6\"$ ?

To find these two numbers, we need to list all of the factors of $\"-16\"$ (the previous product).

Factors of $\"-16\"$ :

1,2,4,8,16

-1,-2,-4,-8,-16

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to $\"-16\"$ .

1*(-16) = -16
2*(-8) = -16
4*(-4) = -16
(-1)*(16) = -16
(-2)*(8) = -16
(-4)*(4) = -16

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-6\"$ :

\n" ); document.write( "

First Number	Second Number	Sum
1	-16	1+(-16)=-15
2	-8	2+(-8)=-6
4	-4	4+(-4)=0
-1	16	-1+16=15
-2	8	-2+8=6
-4	4	-4+4=0

From the table, we can see that the two numbers $\"2\"$ and $\"-8\"$ add to $\"-6\"$ (the middle coefficient).

So the two numbers $\"2\"$ and $\"-8\"$ both multiply to $\"-16\"$ and add to $\"-6\"$

Now replace the middle term $\"-6c\"$ with $\"2c-8c\"$ . Remember, $\"2\"$ and $\"-8\"$ add to $\"-6\"$ . So this shows us that $\"2c-8c=-6c\"$ .

$\"c%5E2%2Bhighlight%282c-8c%29-16\"$ Replace the second term $\"-6c\"$ with $\"2c-8c\"$ .

$\"%28c%5E2%2B2c%29%2B%28-8c-16%29\"$ Group the terms into two pairs.

$\"c%28c%2B2%29%2B%28-8c-16%29\"$ Factor out the GCF $\"c\"$ from the first group.

$\"c%28c%2B2%29-8%28c%2B2%29\"$ Factor out $\"8\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

$\"%28c-8%29%28c%2B2%29\"$ Combine like terms. Or factor out the common term $\"c%2B2\"$

===============================================================

Answer:

So $\"c%5E2-6%2Ac-16\"$ factors to $\"%28c-8%29%28c%2B2%29\"$ .

In other words, $\"c%5E2-6%2Ac-16=%28c-8%29%28c%2B2%29\"$ .

Note: you can check the answer by expanding $\"%28c-8%29%28c%2B2%29\"$ to get $\"c%5E2-6%2Ac-16\"$ or by graphing the original expression and the answer (the two graphs should be identical).

\n" ); document.write( " \n" ); document.write( "

\n" );