document.write( "Question 101242: Factor the following trinomial completely\r
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Algebra.Com's Answer #73738 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)

Looking at the expression $\"x%5E2-11x-60\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-11\"$ , and the last term is $\"-60\"$ .

Now multiply the first coefficient $\"1\"$ by the last term $\"-60\"$ to get $\"%281%29%28-60%29=-60\"$ .

Now the question is: what two whole numbers multiply to $\"-60\"$ (the previous product) and add to the second coefficient $\"-11\"$ ?

To find these two numbers, we need to list all of the factors of $\"-60\"$ (the previous product).

Factors of $\"-60\"$ :

1,2,3,4,5,6,10,12,15,20,30,60

-1,-2,-3,-4,-5,-6,-10,-12,-15,-20,-30,-60

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

These factors pair up and multiply to $\"-60\"$ .

1*(-60) = -60
2*(-30) = -60
3*(-20) = -60
4*(-15) = -60
5*(-12) = -60
6*(-10) = -60
(-1)*(60) = -60
(-2)*(30) = -60
(-3)*(20) = -60
(-4)*(15) = -60
(-5)*(12) = -60
(-6)*(10) = -60

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-11\"$ :

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First Number	Second Number	Sum
1	-60	1+(-60)=-59
2	-30	2+(-30)=-28
3	-20	3+(-20)=-17
4	-15	4+(-15)=-11
5	-12	5+(-12)=-7
6	-10	6+(-10)=-4
-1	60	-1+60=59
-2	30	-2+30=28
-3	20	-3+20=17
-4	15	-4+15=11
-5	12	-5+12=7
-6	10	-6+10=4

From the table, we can see that the two numbers $\"4\"$ and $\"-15\"$ add to $\"-11\"$ (the middle coefficient).

So the two numbers $\"4\"$ and $\"-15\"$ both multiply to $\"-60\"$ and add to $\"-11\"$

Now replace the middle term $\"-11x\"$ with $\"4x-15x\"$ . Remember, $\"4\"$ and $\"-15\"$ add to $\"-11\"$ . So this shows us that $\"4x-15x=-11x\"$ .

$\"x%5E2%2Bhighlight%284x-15x%29-60\"$ Replace the second term $\"-11x\"$ with $\"4x-15x\"$ .

$\"%28x%5E2%2B4x%29%2B%28-15x-60%29\"$ Group the terms into two pairs.

$\"x%28x%2B4%29%2B%28-15x-60%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%28x%2B4%29-15%28x%2B4%29\"$ Factor out $\"15\"$ 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.

$\"%28x-15%29%28x%2B4%29\"$ Combine like terms. Or factor out the common term $\"x%2B4\"$

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Answer:

So $\"x%5E2-11%2Ax-60\"$ factors to $\"%28x-15%29%28x%2B4%29\"$ .

In other words, $\"x%5E2-11%2Ax-60=%28x-15%29%28x%2B4%29\"$ .

Note: you can check the answer by expanding $\"%28x-15%29%28x%2B4%29\"$ to get $\"x%5E2-11%2Ax-60\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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