SOLUTION: i want to know how to factor 48x^3 - 40x^2 - 48x

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Question 653914: i want to know how to factor 48x^3 - 40x^2 - 48x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

48x%5E3-40x%5E2-48x Start with the given expression.


8x%286x%5E2-5x-6%29 Factor out the GCF 8x.


Now let's try to factor the inner expression 6x%5E2-5x-6


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Looking at the expression 6x%5E2-5x-6, we can see that the first coefficient is 6, the second coefficient is -5, and the last term is -6.


Now multiply the first coefficient 6 by the last term -6 to get %286%29%28-6%29=-36.


Now the question is: what two whole numbers multiply to -36 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of -36 (the previous product).


Factors of -36:
1,2,3,4,6,9,12,18,36
-1,-2,-3,-4,-6,-9,-12,-18,-36


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


These factors pair up and multiply to -36.
1*(-36) = -36
2*(-18) = -36
3*(-12) = -36
4*(-9) = -36
6*(-6) = -36
(-1)*(36) = -36
(-2)*(18) = -36
(-3)*(12) = -36
(-4)*(9) = -36
(-6)*(6) = -36

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:


First NumberSecond NumberSum
1-361+(-36)=-35
2-182+(-18)=-16
3-123+(-12)=-9
4-94+(-9)=-5
6-66+(-6)=0
-136-1+36=35
-218-2+18=16
-312-3+12=9
-49-4+9=5
-66-6+6=0



From the table, we can see that the two numbers 4 and -9 add to -5 (the middle coefficient).


So the two numbers 4 and -9 both multiply to -36 and add to -5


Now replace the middle term -5x with 4x-9x. Remember, 4 and -9 add to -5. So this shows us that 4x-9x=-5x.


6x%5E2%2Bhighlight%284x-9x%29-6 Replace the second term -5x with 4x-9x.


%286x%5E2%2B4x%29%2B%28-9x-6%29 Group the terms into two pairs.


2x%283x%2B2%29%2B%28-9x-6%29 Factor out the GCF 2x from the first group.


2x%283x%2B2%29-3%283x%2B2%29 Factor out 3 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.


%282x-3%29%283x%2B2%29 Combine like terms. Or factor out the common term 3x%2B2


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So 8x%286x%5E2-5x-6%29 then factors further to 8x%282x-3%29%283x%2B2%29


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Answer:


So 48x%5E3-40x%5E2-48x completely factors to 8x%282x-3%29%283x%2B2%29.


In other words, 48x%5E3-40x%5E2-48x=8x%282x-3%29%283x%2B2%29.


Note: you can check the answer by expanding 8x%282x-3%29%283x%2B2%29 to get 48x%5E3-40x%5E2-48x or by graphing the original expression and the answer (the two graphs should be identical).

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