document.write( "Question 112193: Completely factor the polynomial 2z^3+14z^2-60z \n" ); document.write( "

Algebra.Com's Answer #81842 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"2z%5E3%2B14z%5E2-60z\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2z%28z%5E2%2B7z-30%29\"$ Factor out the GCF $\"2z\"$ \r
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"z%5E2%2B7z-30\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"7\"$ , and the last term is $\"-30\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-30\"$ to get $\"%281%29%28-30%29=-30\"$ .

Now the question is: what two whole numbers multiply to $\"-30\"$ (the previous product) and add to the second coefficient $\"7\"$ ?

To find these two numbers, we need to list all of the factors of $\"-30\"$ (the previous product).

Factors of $\"-30\"$ :

1,2,3,5,6,10,15,30

-1,-2,-3,-5,-6,-10,-15,-30

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

These factors pair up and multiply to $\"-30\"$ .

1*(-30) = -30
2*(-15) = -30
3*(-10) = -30
5*(-6) = -30
(-1)*(30) = -30
(-2)*(15) = -30
(-3)*(10) = -30
(-5)*(6) = -30

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"7\"$ :

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First Number	Second Number	Sum
1	-30	1+(-30)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

From the table, we can see that the two numbers $\"-3\"$ and $\"10\"$ add to $\"7\"$ (the middle coefficient).

So the two numbers $\"-3\"$ and $\"10\"$ both multiply to $\"-30\"$ and add to $\"7\"$

Now replace the middle term $\"7z\"$ with $\"-3z%2B10z\"$ . Remember, $\"-3\"$ and $\"10\"$ add to $\"7\"$ . So this shows us that $\"-3z%2B10z=7z\"$ .

$\"z%5E2%2Bhighlight%28-3z%2B10z%29-30\"$ Replace the second term $\"7z\"$ with $\"-3z%2B10z\"$ .

$\"%28z%5E2-3z%29%2B%2810z-30%29\"$ Group the terms into two pairs.

$\"z%28z-3%29%2B%2810z-30%29\"$ Factor out the GCF $\"z\"$ from the first group.

$\"z%28z-3%29%2B10%28z-3%29\"$ Factor out $\"10\"$ 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.

$\"%28z%2B10%29%28z-3%29\"$ Combine like terms. Or factor out the common term $\"z-3\"$

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Answer:

So $\"z%5E2%2B7%2Az-30\"$ factors to $\"%28z%2B10%29%28z-3%29\"$ .

In other words, $\"z%5E2%2B7%2Az-30=%28z%2B10%29%28z-3%29\"$ .

Note: you can check the answer by expanding $\"%28z%2B10%29%28z-3%29\"$ to get $\"z%5E2%2B7%2Az-30\"$ 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( " $\"2z%28z%2B10%29%28z-3%29\"$ Now replace $\"z%5E2%2B7z-30\"$ with $\"%28z%2B10%29%28z-3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"2z%5E3%2B14z%5E2-60z\"$ factors to $\"2z%28z%2B10%29%28z-3%29\"$ \n" ); document.write( "

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