document.write( "Question 102391: Factor the polynomial completely. If the polynomial cannot be factored, choose prime\r
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Algebra.Com's Answer #74418 by jim_thompson5910(35256) $\"\"$ $\"About$

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

Looking at the expression $\"7x%5E2-9x-10\"$ , we can see that the first coefficient is $\"7\"$ , the second coefficient is $\"-9\"$ , and the last term is $\"-10\"$ .

Now multiply the first coefficient $\"7\"$ by the last term $\"-10\"$ to get $\"%287%29%28-10%29=-70\"$ .

Now the question is: what two whole numbers multiply to $\"-70\"$ (the previous product) and add to the second coefficient $\"-9\"$ ?

To find these two numbers, we need to list all of the factors of $\"-70\"$ (the previous product).

Factors of $\"-70\"$ :

1,2,5,7,10,14,35,70

-1,-2,-5,-7,-10,-14,-35,-70

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

These factors pair up and multiply to $\"-70\"$ .

1*(-70) = -70
2*(-35) = -70
5*(-14) = -70
7*(-10) = -70
(-1)*(70) = -70
(-2)*(35) = -70
(-5)*(14) = -70
(-7)*(10) = -70

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	-70	1+(-70)=-69
2	-35	2+(-35)=-33
5	-14	5+(-14)=-9
7	-10	7+(-10)=-3
-1	70	-1+70=69
-2	35	-2+35=33
-5	14	-5+14=9
-7	10	-7+10=3

From the table, we can see that the two numbers $\"5\"$ and $\"-14\"$ add to $\"-9\"$ (the middle coefficient).

So the two numbers $\"5\"$ and $\"-14\"$ both multiply to $\"-70\"$ and add to $\"-9\"$

Now replace the middle term $\"-9x\"$ with $\"5x-14x\"$ . Remember, $\"5\"$ and $\"-14\"$ add to $\"-9\"$ . So this shows us that $\"5x-14x=-9x\"$ .

$\"7x%5E2%2Bhighlight%285x-14x%29-10\"$ Replace the second term $\"-9x\"$ with $\"5x-14x\"$ .

$\"%287x%5E2%2B5x%29%2B%28-14x-10%29\"$ Group the terms into two pairs.

$\"x%287x%2B5%29%2B%28-14x-10%29\"$ Factor out the GCF $\"x\"$ from the first group.

$\"x%287x%2B5%29-2%287x%2B5%29\"$ 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.

$\"%28x-2%29%287x%2B5%29\"$ Combine like terms. Or factor out the common term $\"7x%2B5\"$

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Answer:

So $\"7%2Ax%5E2-9%2Ax-10\"$ factors to $\"%28x-2%29%287x%2B5%29\"$ .

In other words, $\"7%2Ax%5E2-9%2Ax-10=%28x-2%29%287x%2B5%29\"$ .

Note: you can check the answer by expanding $\"%28x-2%29%287x%2B5%29\"$ to get $\"7%2Ax%5E2-9%2Ax-10\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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