document.write( "Question 894960: How would I factor \"3k^2-24k-60\" completely? \n" ); document.write( "

Algebra.Com's Answer #542455 by richwmiller(17219) $\"\"$ $\"About$

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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

$\"3%2Ak%5E2-24%2Ak-60\"$ Start with the given expression.

$\"3%28k%5E2-8k-20%29\"$ Factor out the GCF $\"3\"$ .

Now let's try to factor the inner expression $\"k%5E2-8k-20\"$

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Looking at the expression $\"k%5E2-8k-20\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-8\"$ , and the last term is $\"-20\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-20\"$ to get $\"%281%29%28-20%29=-20\"$ .

Now the question is: what two whole numbers multiply to $\"-20\"$ (the previous product) and add to the second coefficient $\"-8\"$ ?

To find these two numbers, we need to list all of the factors of $\"-20\"$ (the previous product).

Factors of $\"-20\"$ :

1,2,4,5,10,20

-1,-2,-4,-5,-10,-20

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

These factors pair up and multiply to $\"-20\"$ .

1*(-20) = -20
2*(-10) = -20
4*(-5) = -20
(-1)*(20) = -20
(-2)*(10) = -20
(-4)*(5) = -20

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-8\"$ :

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First Number	Second Number	Sum
1	-20	1+(-20)=-19
2	-10	2+(-10)=-8
4	-5	4+(-5)=-1
-1	20	-1+20=19
-2	10	-2+10=8
-4	5	-4+5=1

From the table, we can see that the two numbers $\"2\"$ and $\"-10\"$ add to $\"-8\"$ (the middle coefficient).

So the two numbers $\"2\"$ and $\"-10\"$ both multiply to $\"-20\"$ and add to $\"-8\"$

Now replace the middle term $\"-8k\"$ with $\"2k-10k\"$ . Remember, $\"2\"$ and $\"-10\"$ add to $\"-8\"$ . So this shows us that $\"2k-10k=-8k\"$ .

$\"k%5E2%2Bhighlight%282k-10k%29-20\"$ Replace the second term $\"-8k\"$ with $\"2k-10k\"$ .

$\"%28k%5E2%2B2k%29%2B%28-10k-20%29\"$ Group the terms into two pairs.

$\"k%28k%2B2%29%2B%28-10k-20%29\"$ Factor out the GCF $\"k\"$ from the first group.

$\"k%28k%2B2%29-10%28k%2B2%29\"$ 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.

$\"%28k-10%29%28k%2B2%29\"$ Combine like terms. Or factor out the common term $\"k%2B2\"$

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So $\"3%28k%5E2-8k-20%29\"$ then factors further to $\"3%28k-10%29%28k%2B2%29\"$

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Answer:

So $\"3%2Ak%5E2-24%2Ak-60\"$ completely factors to $\"3%28k-10%29%28k%2B2%29\"$ .

In other words, $\"3%2Ak%5E2-24%2Ak-60=3%28k-10%29%28k%2B2%29\"$ .

Note: you can check the answer by expanding $\"3%28k-10%29%28k%2B2%29\"$ to get $\"3%2Ak%5E2-24%2Ak-60\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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