Question 117545: please help me
factor completely with respect to the integers
x^4-13x^2+36
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Looking at  we can see that the first term is  and the last term is  where the coefficients are 1 and 36 respectively.
Now multiply the first coefficient 1 and the last coefficient 36 to get 36. Now what two numbers multiply to 36 and add to the middle coefficient -13? Let's list all of the factors of 36:
Factors of 36:
1,2,3,4,6,9,12,18
-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. This will allow us to find all possible combinations
These factors pair up and multiply to 36
1*36
2*18
3*12
4*9
6*6
(-1)*(-36)
(-2)*(-18)
(-3)*(-12)
(-4)*(-9)
(-6)*(-6)
note: remember two negative numbers multiplied together make a positive number
Now which of these pairs add to -13? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -13
| First Number | Second Number | Sum | | 1 | 36 | 1+36=37 | | 2 | 18 | 2+18=20 | | 3 | 12 | 3+12=15 | | 4 | 9 | 4+9=13 | | 6 | 6 | 6+6=12 | | -1 | -36 | -1+(-36)=-37 | | -2 | -18 | -2+(-18)=-20 | | -3 | -12 | -3+(-12)=-15 | | -4 | -9 | -4+(-9)=-13 | | -6 | -6 | -6+(-6)=-12 |
From this list we can see that -4 and -9 add up to -13 and multiply to 36
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
 Now factor  and using the difference of squares
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Answer:
So completely factors to
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