document.write( "Question 110952: factor completly
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Algebra.Com's Answer #80927 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"6z%5E3-27z%5E2%2B12z\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"3z%282z%5E2-9z%2B4%29\"$ Factor out the GCF $\"3z\"$ \r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"2z%5E2-9z%2B4\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"-9\"$ , and the last term is $\"4\"$ .

Now multiply the first coefficient $\"2\"$ by the last term $\"4\"$ to get $\"%282%29%284%29=8\"$ .

Now the question is: what two whole numbers multiply to $\"8\"$ (the previous product) and add to the second coefficient $\"-9\"$ ?

To find these two numbers, we need to list all of the factors of $\"8\"$ (the previous product).

Factors of $\"8\"$ :

1,2,4,8

-1,-2,-4,-8

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

These factors pair up and multiply to $\"8\"$ .

1*8 = 8
2*4 = 8
(-1)*(-8) = 8
(-2)*(-4) = 8

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-9\"$ :

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First Number	Second Number	Sum
1	8	1+8=9
2	4	2+4=6
-1	-8	-1+(-8)=-9
-2	-4	-2+(-4)=-6

From the table, we can see that the two numbers $\"-1\"$ and $\"-8\"$ add to $\"-9\"$ (the middle coefficient).

So the two numbers $\"-1\"$ and $\"-8\"$ both multiply to $\"8\"$ and add to $\"-9\"$

Now replace the middle term $\"-9z\"$ with $\"-z-8z\"$ . Remember, $\"-1\"$ and $\"-8\"$ add to $\"-9\"$ . So this shows us that $\"-z-8z=-9z\"$ .

$\"2z%5E2%2Bhighlight%28-z-8z%29%2B4\"$ Replace the second term $\"-9z\"$ with $\"-z-8z\"$ .

$\"%282z%5E2-z%29%2B%28-8z%2B4%29\"$ Group the terms into two pairs.

$\"z%282z-1%29%2B%28-8z%2B4%29\"$ Factor out the GCF $\"z\"$ from the first group.

$\"z%282z-1%29-4%282z-1%29\"$ Factor out $\"4\"$ 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.

$\"%28z-4%29%282z-1%29\"$ Combine like terms. Or factor out the common term $\"2z-1\"$

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Answer:

So $\"2%2Az%5E2-9%2Az%2B4\"$ factors to $\"%28z-4%29%282z-1%29\"$ .

In other words, $\"2%2Az%5E2-9%2Az%2B4=%28z-4%29%282z-1%29\"$ .

Note: you can check the answer by expanding $\"%28z-4%29%282z-1%29\"$ to get $\"2%2Az%5E2-9%2Az%2B4\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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\n" ); document.write( "\n" ); document.write( "So $\"6z%5E3-27z%5E2%2B12z\"$ factors to $\"3z%28z-4%29%282z-1%29%29\"$ \n" ); document.write( "

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