SOLUTION: Factor....Need some help x^4-13x^2+36

Question 168695: Factor....Need some help
x^4-13x^2+36
Found 3 solutions by jim_thompson5910, Alan3354, ankor@dixie-net.com:
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

So this also means that factors to (since is equivalent to )

So factors to

-----------------------------

Start with the given factorization

Factor to get (by use of the difference of squares)

Factor to get (by use of the difference of squares)

------------------------------------------------------------

Answer:

So completely factors to

Note: the order of the factors does not matter.
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
Factor....Need some help
x^4-13x^2+36
----------------
It will be (x^2 + ?)*(x^2 + ?)
The ?'s product will be 36, and they add up to -13.
Since it's +36, they have to have the same sign, both Pos or both Neg.
Since they add to -13, they'll have to be both negative.
Consider the factors of 36:
1*36
2*18
3*12
4*9, and
6*6
They're the same sign, so that add to 13. If the were not the same sign, you would look at the differences.
Only 4 and 9 add to 13, so they're it.
It's (x^2 - 4)*(x^2 - 9)

Answer by ankor@dixie-net.com(22740)   (Show Source): You can put this solution on YOUR website!
x^4 - 13x^2 + 36
Factor this like an ordinary quadratic except you have:
:
(x^2 - 9)(x^2 - 4); (If you FOIL this you will get the original expression)
:
Note that each factor is the "difference of squares" and can be further factored
(x^2 - 9) = (x - 3)(x + 3)
and
(x^2 - 4) = (x - 2)(x + 2)

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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