document.write( "Question 111838: Factor -3s^2-10s+8 \n" ); document.write( "

Algebra.Com's Answer #81570 by jim_thompson5910(35256) $\"\"$ $\"About$

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

$\"-3%2As%5E2-10%2As%2B8\"$ Start with the given expression.

$\"-%283s%5E2%2B10s-8%29\"$ Factor out the GCF $\"-1\"$ .

Now let's try to factor the inner expression $\"3s%5E2%2B10s-8\"$

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Looking at the expression $\"3s%5E2%2B10s-8\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"10\"$ , and the last term is $\"-8\"$ .

Now multiply the first coefficient $\"3\"$ by the last term $\"-8\"$ to get $\"%283%29%28-8%29=-24\"$ .

Now the question is: what two whole numbers multiply to $\"-24\"$ (the previous product) and add to the second coefficient $\"10\"$ ?

To find these two numbers, we need to list all of the factors of $\"-24\"$ (the previous product).

Factors of $\"-24\"$ :

1,2,3,4,6,8,12,24

-1,-2,-3,-4,-6,-8,-12,-24

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

These factors pair up and multiply to $\"-24\"$ .

1*(-24) = -24
2*(-12) = -24
3*(-8) = -24
4*(-6) = -24
(-1)*(24) = -24
(-2)*(12) = -24
(-3)*(8) = -24
(-4)*(6) = -24

Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"10\"$ :

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First Number	Second Number	Sum
1	-24	1+(-24)=-23
2	-12	2+(-12)=-10
3	-8	3+(-8)=-5
4	-6	4+(-6)=-2
-1	24	-1+24=23
-2	12	-2+12=10
-3	8	-3+8=5
-4	6	-4+6=2

From the table, we can see that the two numbers $\"-2\"$ and $\"12\"$ add to $\"10\"$ (the middle coefficient).

So the two numbers $\"-2\"$ and $\"12\"$ both multiply to $\"-24\"$ and add to $\"10\"$

Now replace the middle term $\"10s\"$ with $\"-2s%2B12s\"$ . Remember, $\"-2\"$ and $\"12\"$ add to $\"10\"$ . So this shows us that $\"-2s%2B12s=10s\"$ .

$\"3s%5E2%2Bhighlight%28-2s%2B12s%29-8\"$ Replace the second term $\"10s\"$ with $\"-2s%2B12s\"$ .

$\"%283s%5E2-2s%29%2B%2812s-8%29\"$ Group the terms into two pairs.

$\"s%283s-2%29%2B%2812s-8%29\"$ Factor out the GCF $\"s\"$ from the first group.

$\"s%283s-2%29%2B4%283s-2%29\"$ 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.

$\"%28s%2B4%29%283s-2%29\"$ Combine like terms. Or factor out the common term $\"3s-2\"$

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So $\"-1%283s%5E2%2B10s-8%29\"$ then factors further to $\"-%28s%2B4%29%283s-2%29\"$

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Answer:

So $\"-3%2As%5E2-10%2As%2B8\"$ completely factors to $\"-%28s%2B4%29%283s-2%29\"$ .

In other words, $\"-3%2As%5E2-10%2As%2B8=-%28s%2B4%29%283s-2%29\"$ .

Note: you can check the answer by expanding $\"-%28s%2B4%29%283s-2%29\"$ to get $\"-3%2As%5E2-10%2As%2B8\"$ or by graphing the original expression and the answer (the two graphs should be identical).

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