SOLUTION: factor completely (-x^3 +x^2 +42x)

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Question 338529: factor completely
(-x^3 +x^2 +42x)
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

-x%5E3%2Bx%5E2%2B42x Start with the given expression.


-x%28x%5E2-x-42%29 Factor out the GCF -x.


Now let's try to factor the inner expression x%5E2-x-42


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Looking at the expression x%5E2-x-42, we can see that the first coefficient is 1, the second coefficient is -1, and the last term is -42.


Now multiply the first coefficient 1 by the last term -42 to get %281%29%28-42%29=-42.


Now the question is: what two whole numbers multiply to -42 (the previous product) and add to the second coefficient -1?


To find these two numbers, we need to list all of the factors of -42 (the previous product).


Factors of -42:
1,2,3,6,7,14,21,42
-1,-2,-3,-6,-7,-14,-21,-42


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


These factors pair up and multiply to -42.
1*(-42) = -42
2*(-21) = -42
3*(-14) = -42
6*(-7) = -42
(-1)*(42) = -42
(-2)*(21) = -42
(-3)*(14) = -42
(-6)*(7) = -42

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -1:


First NumberSecond NumberSum
1-421+(-42)=-41
2-212+(-21)=-19
3-143+(-14)=-11
6-76+(-7)=-1
-142-1+42=41
-221-2+21=19
-314-3+14=11
-67-6+7=1



From the table, we can see that the two numbers 6 and -7 add to -1 (the middle coefficient).


So the two numbers 6 and -7 both multiply to -42 and add to -1


Now replace the middle term -1x with 6x-7x. Remember, 6 and -7 add to -1. So this shows us that 6x-7x=-1x.


x%5E2%2Bhighlight%286x-7x%29-42 Replace the second term -1x with 6x-7x.


%28x%5E2%2B6x%29%2B%28-7x-42%29 Group the terms into two pairs.


x%28x%2B6%29%2B%28-7x-42%29 Factor out the GCF x from the first group.


x%28x%2B6%29-7%28x%2B6%29 Factor out 7 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-7%29%28x%2B6%29 Combine like terms. Or factor out the common term x%2B6


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So -1x%28x%5E2-x-42%29 then factors further to -%28x-7%29%28x%2B6%29


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Answer:


So -x%5E3%2Bx%5E2%2B42x completely factors to -%28x-7%29%28x%2B6%29.


In other words, -x%5E3%2Bx%5E2%2B42x=-x%28x-7%29%28x%2B6%29.


Note: you can check the answer by expanding -x%28x-7%29%28x%2B6%29 to get -x%5E3%2Bx%5E2%2B42x or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website.

Jim