SOLUTION: How to factor 2x^3-x^2-10x+5 by grouping

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: How to factor 2x^3-x^2-10x+5 by grouping      Log On


   

Question 816155: How to factor 2x^3-x^2-10x+5 by grouping
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
2x%5E3-x%5E2-10x%2B5
I find that factoring by grouping is easier if you
  • Change any subtractions into equivalent additions.
  • When factoring out the greatest common factor (GCF) from each group, consider factoring out the negative of the GCF from just one of the groups.
Let's see this on your expression. First, change the subtractions:
2x%5E3%2B%28-x%5E2%29%2B%28-10x%29%2B5
Now we group:
%282x%5E3%2B%28-x%5E2%29%29%2B%28%28-10x%29%2B5%29
Next we factor out the GCF from each group. The GCF of the first group is x%5E2. The GCF of the second group is 5. Factoring these out we get:
x%5E2%7B2x%2B%28-1%29%29%2B5%28%28-2x%29%2B1%29
As we can see the "non-GCF" factors, (2x+(-1)) and (-2x+1) are not the same. So it appears that factoring by grouping is not working for us. But these two factors are exact opposites of each other. This is when factoring our the negative of the GCF can be very helpful. Instead of factoring 5 out of the second group of 2x%5E3%2B%28-x%5E2%29%29%2B%28%28-10x%29%2B5%29 we will factor out -5:
x%5E2%282x%2B%28-1%29%29%2B%28-5%29%282x%2B%28-1%29%29
Now the "non-GCF" factors are the same and we can proceed. Next we factor out the GCF of the two groups, (2x+(-1)):
%282x%2B%28-1%29%29%28x%5E2%2B%28-5%29%29
Since neither of these factors will factor further, we are finished factoring. At this point additions of negatives may be changed back to subtractions if you prefer. So the answer is either
%282x%2B%28-1%29%29%28x%5E2%2B%28-5%29%29
or
%282x-1%29%28x%5E2-5%29