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Question 928275: Please show the steps to factor completely
6x^3 +5x^2 -4x
Found 2 solutions by jim_thompson5910, KMST:
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
 Start with the given expression.
 Factor out the GCF  .
Now let's try to factor the inner expression 
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Looking at the expression , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to  (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
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 .
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 :
| 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  and  add to (the middle coefficient).
So the two numbers and both multiply to and add to
Now replace the middle term  with  . Remember, and add to . So this shows us that  .
 Replace the second term with .
 Group the terms into two pairs.
 Factor out the GCF  from the first group.
 Factor out  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.
 Combine like terms. Or factor out the common term 
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So then factors further to 
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Answer:
So completely factors to .
In other words,  .
Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).
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Jim
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Answer by KMST(5328)  (Show Source):
You can put this solution on YOUR website! 
STEP 1: Look for common factors and take out common factors if you find any:
Always look for common factors, at the beginning and in further steps.
STEP 2: If the highest degree of your factors is 2,
factoring further )factoring the factors) may be possible, and even easy.
If  can be factored further,
we will get to the factoring by looking for factors of  ,
the product of the first and last coefficients.
We need a pair of factors that multiply to yield and add up to ,
the middle coefficient.
It turns out that  and  ,
so and are the factors we are looking for.
We re-write  as , and then "factor by grouping" :
 .
There we took as a common factor out of the group  ,
and we took common factor out of group  .
Next we took common factor  out of the whole thing to get
 .
STEP #3: Put it all together:
 .
NOTE: If you cannot figure out how to factor a quadratic factor/polynomial,
like , there is a sort of cheat.
You could try making it equal to zero,
and solving the resulting quadratic equation by using the quadratic formula.
If there is no solution, the quadratic factor/polynomial cannot be factored.
If there is a solution
 --> -->
The solutions to the equation, subtracted from are factors.
In this case,  and  are factors of .
Since the denominators make them look ugly, we multiply times the denominators to find the factors
 and  .
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