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

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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) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at x%5E4-13x%5E2%2B36 we can see that the first term is 1x%5E4 and the last term is 36 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 NumberSecond NumberSum
1361+36=37
2182+18=20
3123+12=15
494+9=13
666+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 1x%5E4-13x%5E2%2B36, replace -13x%5E2 with -4x%5E2%2B-9x%5E2 (notice -4x%5E2%2B-9x%5E2 adds up to -13x%5E2. So it is equivalent to -13x%5E2)

x%5E4%2Bhighlight%28-4x%5E2%2B-9x%5E2%29%2B36


Now let's factor x%5E4-4x%5E2-9x%5E2%2B36 by grouping:


%28x%5E4-4x%5E2%29%2B%28-9x%5E2%2B36%29 Group like terms


x%5E2%28x%5E2-4%29-9%28x%5E2-4%29 Factor out the GCF of x%5E2 out of the first group. Factor out the GCF of -9 out of the second group


%28x%5E2-9%29%28x%5E2-4%29 Since we have a common term of x%5E2-4, we can combine like terms

So x%5E4-4x%5E2-9x%5E2%2B36 factors to %28x%5E2-9%29%28x%5E2-4%29


So this also means that x%5E4-13x%5E2%2B36 factors to %28x%5E2-9%29%28x%5E2-4%29 (since x%5E4-13x%5E2%2B36 is equivalent to x%5E4-4x%5E2-9x%5E2%2B36)



So x%5E4-13x%5E2%2B36 factors to %28x%5E2-9%29%28x%5E2-4%29

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


%28x%5E2-9%29%28x%5E2-4%29 Start with the given factorization


%28x%2B3%29%28x-3%29%28x%5E2-4%29 Factor x%5E2-9 to get %28x%2B3%29%28x-3%29 (by use of the difference of squares)


%28x%2B3%29%28x-3%29%28x%2B2%29%28x-2%29 Factor x%5E2-4 to get %28x%2B2%29%28x-2%29 (by use of the difference of squares)



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Answer:


So x%5E4-13x%5E2%2B36 completely factors to %28x%2B3%29%28x-3%29%28x%2B2%29%28x-2%29


Note: the order of the factors does not matter.

Answer by Alan3354(69443) About Me  (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) About Me  (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)