SOLUTION: Factor if possible x^3+8x^2+12x thanks.

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Question 725324: Factor if possible
x^3+8x^2+12x
thanks.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E3%2B8x%5E2%2B12x Start with the given expression.


x%28x%5E2%2B8x%2B12%29 Factor out the GCF x.


Now let's try to factor the inner expression x%5E2%2B8x%2B12


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Looking at the expression x%5E2%2B8x%2B12, we can see that the first coefficient is 1, the second coefficient is 8, and the last term is 12.


Now multiply the first coefficient 1 by the last term 12 to get %281%29%2812%29=12.


Now the question is: what two whole numbers multiply to 12 (the previous product) and add to the second coefficient 8?


To find these two numbers, we need to list all of the factors of 12 (the previous product).


Factors of 12:
1,2,3,4,6,12
-1,-2,-3,-4,-6,-12


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


These factors pair up and multiply to 12.
1*12 = 12
2*6 = 12
3*4 = 12
(-1)*(-12) = 12
(-2)*(-6) = 12
(-3)*(-4) = 12

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


First NumberSecond NumberSum
1121+12=13
262+6=8
343+4=7
-1-12-1+(-12)=-13
-2-6-2+(-6)=-8
-3-4-3+(-4)=-7



From the table, we can see that the two numbers 2 and 6 add to 8 (the middle coefficient).


So the two numbers 2 and 6 both multiply to 12 and add to 8


Now replace the middle term 8x with 2x%2B6x. Remember, 2 and 6 add to 8. So this shows us that 2x%2B6x=8x.


x%5E2%2Bhighlight%282x%2B6x%29%2B12 Replace the second term 8x with 2x%2B6x.


%28x%5E2%2B2x%29%2B%286x%2B12%29 Group the terms into two pairs.


x%28x%2B2%29%2B%286x%2B12%29 Factor out the GCF x from the first group.


x%28x%2B2%29%2B6%28x%2B2%29 Factor out 6 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%2B6%29%28x%2B2%29 Combine like terms. Or factor out the common term x%2B2


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So x%28x%5E2%2B8x%2B12%29 then factors further to x%28x%2B6%29%28x%2B2%29


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Answer:


So x%5E3%2B8x%5E2%2B12x completely factors to x%28x%2B6%29%28x%2B2%29.


In other words, x%5E3%2B8x%5E2%2B12x=x%28x%2B6%29%28x%2B2%29.


Note: you can check the answer by expanding x%28x%2B6%29%28x%2B2%29 to get x%5E3%2B8x%5E2%2B12x or by graphing the original expression and the answer (the two graphs should be identical).