document.write( "Question 297591: Factor by grouping\r
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Algebra.Com's Answer #214271 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
For more help with factoring, check out this factoring solver.\r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"14b%5E2-55b%2B21\"$ , we can see that the first coefficient is $\"14\"$ , the second coefficient is $\"-55\"$ , and the last term is $\"21\"$ .

Now multiply the first coefficient $\"14\"$ by the last term $\"21\"$ to get $\"%2814%29%2821%29=294\"$ .

Now the question is: what two whole numbers multiply to $\"294\"$ (the previous product) and add to the second coefficient $\"-55\"$ ?

To find these two numbers, we need to list all of the factors of $\"294\"$ (the previous product).

Factors of $\"294\"$ :

1,2,3,6,7,14,21,42,49,98,147,294

-1,-2,-3,-6,-7,-14,-21,-42,-49,-98,-147,-294

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

These factors pair up and multiply to $\"294\"$ .

1*294 = 294
2*147 = 294
3*98 = 294
6*49 = 294
7*42 = 294
14*21 = 294
(-1)*(-294) = 294
(-2)*(-147) = 294
(-3)*(-98) = 294
(-6)*(-49) = 294
(-7)*(-42) = 294
(-14)*(-21) = 294

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-55\"$ :

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First Number	Second Number	Sum
1	294	1+294=295
2	147	2+147=149
3	98	3+98=101
6	49	6+49=55
7	42	7+42=49
14	21	14+21=35
-1	-294	-1+(-294)=-295
-2	-147	-2+(-147)=-149
-3	-98	-3+(-98)=-101
-6	-49	-6+(-49)=-55
-7	-42	-7+(-42)=-49
-14	-21	-14+(-21)=-35

From the table, we can see that the two numbers $\"-6\"$ and $\"-49\"$ add to $\"-55\"$ (the middle coefficient).

So the two numbers $\"-6\"$ and $\"-49\"$ both multiply to $\"294\"$ and add to $\"-55\"$

Now replace the middle term $\"-55b\"$ with $\"-6b-49b\"$ . Remember, $\"-6\"$ and $\"-49\"$ add to $\"-55\"$ . So this shows us that $\"-6b-49b=-55b\"$ .

$\"14b%5E2%2Bhighlight%28-6b-49b%29%2B21\"$ Replace the second term $\"-55b\"$ with $\"-6b-49b\"$ .

$\"%2814b%5E2-6b%29%2B%28-49b%2B21%29\"$ Group the terms into two pairs.

$\"2b%287b-3%29%2B%28-49b%2B21%29\"$ Factor out the GCF $\"2b\"$ from the first group.

$\"2b%287b-3%29-7%287b-3%29\"$ 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.

$\"%282b-7%29%287b-3%29\"$ Combine like terms. Or factor out the common term $\"7b-3\"$

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Answer:

So $\"14%2Ab%5E2-55%2Ab%2B21\"$ factors to $\"%282b-7%29%287b-3%29\"$ .

In other words, $\"14%2Ab%5E2-55%2Ab%2B21=%282b-7%29%287b-3%29\"$ .

Note: you can check the answer by expanding $\"%282b-7%29%287b-3%29\"$ to get $\"14%2Ab%5E2-55%2Ab%2B21\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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