document.write( "Question 260820: I need to factor 25w^2 - 121 completely. \r
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Algebra.Com's Answer #192138 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
Here is how to factor it
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"25w%5E2%2B0w-121\"$ , we can see that the first coefficient is $\"25\"$ , the second coefficient is $\"0\"$ , and the last term is $\"-121\"$ .

Now multiply the first coefficient $\"25\"$ by the last term $\"-121\"$ to get $\"%2825%29%28-121%29=-3025\"$ .

Now the question is: what two whole numbers multiply to $\"-3025\"$ (the previous product) and add to the second coefficient $\"0\"$ ?

To find these two numbers, we need to list all of the factors of $\"-3025\"$ (the previous product).

Factors of $\"-3025\"$ :

1,5,11,25,55,121,275,605,3025

-1,-5,-11,-25,-55,-121,-275,-605,-3025

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

These factors pair up and multiply to $\"-3025\"$ .

1*(-3025) = -3025
5*(-605) = -3025
11*(-275) = -3025
25*(-121) = -3025
55*(-55) = -3025
(-1)*(3025) = -3025
(-5)*(605) = -3025
(-11)*(275) = -3025
(-25)*(121) = -3025
(-55)*(55) = -3025

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"0\"$ :

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First Number	Second Number	Sum
1	-3025	1+(-3025)=-3024
5	-605	5+(-605)=-600
11	-275	11+(-275)=-264
25	-121	25+(-121)=-96
55	-55	55+(-55)=0
-1	3025	-1+3025=3024
-5	605	-5+605=600
-11	275	-11+275=264
-25	121	-25+121=96
-55	55	-55+55=0

From the table, we can see that the two numbers $\"-55\"$ and $\"55\"$ add to $\"0\"$ (the middle coefficient).

So the two numbers $\"-55\"$ and $\"55\"$ both multiply to $\"-3025\"$ and add to $\"0\"$

Now replace the middle term $\"0w\"$ with $\"-55w%2B55w\"$ . Remember, $\"-55\"$ and $\"55\"$ add to $\"0\"$ . So this shows us that $\"-55w%2B55w=0w\"$ .

$\"25w%5E2%2Bhighlight%28-55w%2B55w%29-121\"$ Replace the second term $\"0w\"$ with $\"-55w%2B55w\"$ .

$\"%2825w%5E2-55w%29%2B%2855w-121%29\"$ Group the terms into two pairs.

$\"5w%285w-11%29%2B%2855w-121%29\"$ Factor out the GCF $\"5w\"$ from the first group.

$\"5w%285w-11%29%2B11%285w-11%29\"$ Factor out $\"11\"$ 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.

$\"%285w%2B11%29%285w-11%29\"$ Combine like terms. Or factor out the common term $\"5w-11\"$

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Answer:

So $\"25%2Aw%5E2%2B0%2Aw-121\"$ factors to $\"%285w%2B11%29%285w-11%29\"$ .

In other words, $\"25%2Aw%5E2%2B0%2Aw-121=%285w%2B11%29%285w-11%29\"$ .

Note: you can check the answer by expanding $\"%285w%2B11%29%285w-11%29\"$ to get $\"25%2Aw%5E2%2B0%2Aw-121\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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