SOLUTION: Please help with step by step solution: 8x^3-10x^2-3x

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Question 922574: Please help with step by step solution:
8x^3-10x^2-3x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor this.


8x%5E3-10x%5E2-3x Start with the given expression.


x%288x%5E2-10x-3%29 Factor out the GCF x.


Now let's try to factor the inner expression 8x%5E2-10x-3


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Looking at the expression 8x%5E2-10x-3, we can see that the first coefficient is 8, the second coefficient is -10, and the last term is -3.


Now multiply the first coefficient 8 by the last term -3 to get %288%29%28-3%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:


First NumberSecond NumberSum
1-241+(-24)=-23
2-122+(-12)=-10
3-83+(-8)=-5
4-64+(-6)=-2
-124-1+24=23
-212-2+12=10
-38-3+8=5
-46-4+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 -10x with 2x-12x. Remember, 2 and -12 add to -10. So this shows us that 2x-12x=-10x.


8x%5E2%2Bhighlight%282x-12x%29-3 Replace the second term -10x with 2x-12x.


%288x%5E2%2B2x%29%2B%28-12x-3%29 Group the terms into two pairs.


2x%284x%2B1%29%2B%28-12x-3%29 Factor out the GCF 2x from the first group.


2x%284x%2B1%29-3%284x%2B1%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%284x%2B1%29 Combine like terms. Or factor out the common term 4x%2B1


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So x%288x%5E2-10x-3%29 then factors further to x%282x-3%29%284x%2B1%29


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Answer:


So 8x%5E3-10x%5E2-3x completely factors to x%282x-3%29%284x%2B1%29.


In other words, 8x%5E3-10x%5E2-3x=x%282x-3%29%284x%2B1%29.


Note: you can check the answer by expanding x%282x-3%29%284x%2B1%29 to get 8x%5E3-10x%5E2-3x or by graphing the original expression and the answer (the two graphs should be identical).

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Let me know if that helps or not. Thanks.

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

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