document.write( "Question 259024: Can you help me factor the polynomial 3a^2+a-14 \n" ); document.write( "

Algebra.Com's Answer #190707 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
(a-2)*(3a+7)
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"3a%5E2%2Ba-14\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"1\"$ , and the last term is $\"-14\"$ .

Now multiply the first coefficient $\"3\"$ by the last term $\"-14\"$ to get $\"%283%29%28-14%29=-42\"$ .

Now the question is: what two whole numbers multiply to $\"-42\"$ (the previous product) and add to the second coefficient $\"1\"$ ?

To find these two numbers, we need to list all of the factors of $\"-42\"$ (the previous product).

Factors of $\"-42\"$ :

1,2,3,6,7,14,21,42

-1,-2,-3,-6,-7,-14,-21,-42

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

These factors pair up and multiply to $\"-42\"$ .

1*(-42) = -42
2*(-21) = -42
3*(-14) = -42
6*(-7) = -42
(-1)*(42) = -42
(-2)*(21) = -42
(-3)*(14) = -42
(-6)*(7) = -42

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"1\"$ :

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First Number	Second Number	Sum
1	-42	1+(-42)=-41
2	-21	2+(-21)=-19
3	-14	3+(-14)=-11
6	-7	6+(-7)=-1
-1	42	-1+42=41
-2	21	-2+21=19
-3	14	-3+14=11
-6	7	-6+7=1

From the table, we can see that the two numbers $\"-6\"$ and $\"7\"$ add to $\"1\"$ (the middle coefficient).

So the two numbers $\"-6\"$ and $\"7\"$ both multiply to $\"-42\"$ and add to $\"1\"$

Now replace the middle term $\"1a\"$ with $\"-6a%2B7a\"$ . Remember, $\"-6\"$ and $\"7\"$ add to $\"1\"$ . So this shows us that $\"-6a%2B7a=1a\"$ .

$\"3a%5E2%2Bhighlight%28-6a%2B7a%29-14\"$ Replace the second term $\"1a\"$ with $\"-6a%2B7a\"$ .

$\"%283a%5E2-6a%29%2B%287a-14%29\"$ Group the terms into two pairs.

$\"3a%28a-2%29%2B%287a-14%29\"$ Factor out the GCF $\"3a\"$ from the first group.

$\"3a%28a-2%29%2B7%28a-2%29\"$ 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.

$\"%283a%2B7%29%28a-2%29\"$ Combine like terms. Or factor out the common term $\"a-2\"$

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Answer:

So $\"3%2Aa%5E2%2Ba-14\"$ factors to $\"%283a%2B7%29%28a-2%29\"$ .

In other words, $\"3%2Aa%5E2%2Ba-14=%283a%2B7%29%28a-2%29\"$ .

Note: you can check the answer by expanding $\"%283a%2B7%29%28a-2%29\"$ to get $\"3%2Aa%5E2%2Ba-14\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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