document.write( "Question 889244: What are the missing dimensions of a solid rectangle if the volume is x^3+5x^2+8x+4 and the height is x+2?\r
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Algebra.Com's Answer #538015 by richwmiller(17219) $\"\"$ $\"About$

You can put this solution on YOUR website!
x^3+5x^2+8 x+4 = (x^2+3x+2) × (x+2)
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2%2B3x%2B2\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"3\"$ , and the last term is $\"2\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"2\"$ to get $\"%281%29%282%29=2\"$ .

Now the question is: what two whole numbers multiply to $\"2\"$ (the previous product) and add to the second coefficient $\"3\"$ ?

To find these two numbers, we need to list all of the factors of $\"2\"$ (the previous product).

Factors of $\"2\"$ :

1,2

-1,-2

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

These factors pair up and multiply to $\"2\"$ .

1*2 = 2
(-1)*(-2) = 2

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"3\"$ :

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First Number	Second Number	Sum
1	2	1+2=3
-1	-2	-1+(-2)=-3

From the table, we can see that the two numbers $\"1\"$ and $\"2\"$ add to $\"3\"$ (the middle coefficient).

So the two numbers $\"1\"$ and $\"2\"$ both multiply to $\"2\"$ and add to $\"3\"$

Now replace the middle term $\"3x\"$ with $\"x%2B2x\"$ . Remember, $\"1\"$ and $\"2\"$ add to $\"3\"$ . So this shows us that $\"x%2B2x=3x\"$ .

$\"x%5E2%2Bhighlight%28x%2B2x%29%2B2\"$ Replace the second term $\"3x\"$ with $\"x%2B2x\"$ .

$\"%28x%5E2%2Bx%29%2B%282x%2B2%29\"$ Group the terms into two pairs.

$\"x%28x%2B1%29%2B%282x%2B2%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x%2B1%29%2B2%28x%2B1%29\"$ Factor out $\"2\"$ 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.

$\"%28x%2B2%29%28x%2B1%29\"$ Combine like terms. Or factor out the common term $\"x%2B1\"$

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Answer:

So $\"x%5E2%2B3%2Ax%2B2\"$ factors to $\"%28x%2B2%29%28x%2B1%29\"$ .

In other words, $\"x%5E2%2B3%2Ax%2B2=%28x%2B2%29%28x%2B1%29\"$ .

Note: you can check the answer by expanding $\"%28x%2B2%29%28x%2B1%29\"$ to get $\"x%5E2%2B3%2Ax%2B2\"$ or by graphing the original expression and the answer (the two graphs should be identical).

\n" ); document.write( "\n" ); document.write( "so the three dimensions are
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