Question 148119: Factor: (2x^(2/3)) - (5x^(1/3)) - 3
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!  Start with the given expression.
Let  . This means that ^{2}=x^{\frac{2}{3}})
 Replace  with  . Replace  with 
Looking at  we can see that the first term is  and the last term is  where the coefficients are 2 and -3 respectively.
Now multiply the first coefficient 2 and the last coefficient -3 to get -6. Now what two numbers multiply to -6 and add to the middle coefficient -5? Let's list all of the factors of -6:
Factors of -6:
1,2,3,6
-1,-2,-3,-6 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to -6
(1)*(-6)
(2)*(-3)
(-1)*(6)
(-2)*(3)
note: remember, the product of a negative and a positive number is a negative number
Now which of these pairs add to -5? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -5
| First Number | Second Number | Sum | | 1 | -6 | 1+(-6)=-5 | | 2 | -3 | 2+(-3)=-1 | | -1 | 6 | -1+6=5 | | -2 | 3 | -2+3=1 |
From this list we can see that 1 and -6 add up to -5 and multiply to -6
Now looking at the expression , replace  with  (notice adds up to . So it is equivalent to )
Now let's factor  by grouping:
 Group like terms
 Factor out the GCF of  out of the first group. Factor out the GCF of out of the second group
 Since we have a common term of  , we can combine like terms
So factors to
\left(2x^{\frac{1}{3}}+1\right)) Now replace "z" with
So factors to
In other words, \left(2x^{\frac{1}{3}}+1\right)) where every variable is positive.
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