document.write( "Question 110944: Factor the trinomial.\r
\n" ); document.write( "\n" ); document.write( "b 2 - 5 b -14 \r
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #80918 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\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 $\"b%5E2-5b-14\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-5\"$ , and the last term is $\"-14\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-14\"$ to get $\"%281%29%28-14%29=-14\"$ .

Now the question is: what two whole numbers multiply to $\"-14\"$ (the previous product) and add to the second coefficient $\"-5\"$ ?

To find these two numbers, we need to list all of the factors of $\"-14\"$ (the previous product).

Factors of $\"-14\"$ :

1,2,7,14

-1,-2,-7,-14

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

These factors pair up and multiply to $\"-14\"$ .

1*(-14) = -14
2*(-7) = -14
(-1)*(14) = -14
(-2)*(7) = -14

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-5\"$ :

\n" ); document.write( "

First Number	Second Number	Sum
1	-14	1+(-14)=-13
2	-7	2+(-7)=-5
-1	14	-1+14=13
-2	7	-2+7=5

From the table, we can see that the two numbers $\"2\"$ and $\"-7\"$ add to $\"-5\"$ (the middle coefficient).

So the two numbers $\"2\"$ and $\"-7\"$ both multiply to $\"-14\"$ and add to $\"-5\"$

Now replace the middle term $\"-5b\"$ with $\"2b-7b\"$ . Remember, $\"2\"$ and $\"-7\"$ add to $\"-5\"$ . So this shows us that $\"2b-7b=-5b\"$ .

$\"b%5E2%2Bhighlight%282b-7b%29-14\"$ Replace the second term $\"-5b\"$ with $\"2b-7b\"$ .

$\"%28b%5E2%2B2b%29%2B%28-7b-14%29\"$ Group the terms into two pairs.

$\"b%28b%2B2%29%2B%28-7b-14%29\"$ Factor out the GCF $\"b\"$ from the first group.

$\"b%28b%2B2%29-7%28b%2B2%29\"$ Factor out $\"7\"$ 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.

$\"%28b-7%29%28b%2B2%29\"$ Combine like terms. Or factor out the common term $\"b%2B2\"$

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

Answer:

So $\"b%5E2-5%2Ab-14\"$ factors to $\"%28b-7%29%28b%2B2%29\"$ .

In other words, $\"b%5E2-5%2Ab-14=%28b-7%29%28b%2B2%29\"$ .

Note: you can check the answer by expanding $\"%28b-7%29%28b%2B2%29\"$ to get $\"b%5E2-5%2Ab-14\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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

\n" );