Question

Factorize
\(x^4-6x^2+1\)

Answer 1

let y = x^2, first

Answer 2

doesn't help much.. tried it.
what i got was
\((x^2-3+2\sqrt{2}) (x^2-3-2\sqrt {2})\)
but that's not required answer.

Answer 3

\[(x^4-2x^2+1)-4x^2\rightarrow (x^2-1)^2-4x^2\rightarrow (x^2-1-2x)(x^2-1+2x)\]

Answer 4

Thanks.
can you help me with this one too?
\(2x^4-5x^3+x^2-5x+2\)

Answer 5

just give a second , it looks like I have to use trial and error

Answer 6

Ok!

Answer 7

this is a tricky one
the factors are probably 2 quadratics of the form

(2x^2 + ax + 1)(x^2 + bx + 2)

Answer 8

a certain amount of trial and error is required

Answer 9

how do you get that?
Basically, what method do you apply while solving this?

Answer 10

try to add and subtract values in the function until you have it on form that enables you to factorize

Answer 11

can you double check the question, I am sort of getting disgusting numbers :\

Answer 12

yeah, its correct!

Answer 13

okay just give me few hours cause I'm kinda bussy right now and I'm afraid that's why those numbers appeared.

Answer 14

ah ok!

Answer 15

\((-6)^{2} - 4(1)(1) = 36 - 4 = 32\) Why do you think it can be factored?

Answer 16

...unless, of course, you mean with something besides integers or rational numbers.

Answer 17

of course, it can be factorized..

Answer 18

I can say that I agree with @tkhunny , this function cannot be factored with the normal ways we that we usually use ,I think Ferrari's method can work with your function, but not possibly to be an exam question in non-advanced level!
However, I believe in similar type of questions you must be given at least one factor of the polynomial to start with..

Answer 19

i finally managed to factor mainly by trial and error

(2x^2 + x + 2)(x^2 - 3x + 1)

the terms in x^2 and the numbers had to be as above (though i guessed the wrong combination on my first try) the x and -3x were obtained by trial and error

Answer 20

that's really interesting @cwrw238
but I still wondering about the way you got the factors, did you only use trial and error, or you just assumed that the function f(x) might be the product for (2x^2+ax+2)(x^2+bx+1) and solved for a and b by equalizing the constants with each other ?

Answer 21

i assumed that the factors would be either (2x^2 + ax + 1)(x^2 + bx + 2) or
(2x^2 + ax + 2)(x^2 + bx + 1)
i didn't get anywhere with the first choice
the term in x, -5x^3 , is formed from 2x^2 * bx and ax * x^2 so
2b + a = -5
and formed other equations for terms in x^2 and x and solved them
they weren't consistent for the first guess but were for the second
so it was part logic and part trial and error

Answer 22

I really appreciate your effort @cwrw238 this was so professional, but I think the problem in this method that it needs a little experience, which means it doesn't depend on a sequence of steps , so I suggest that our friend @ujjwal starts with easier functions and then automatically he will be able to factorize harder ones. and if anyone is concerned, you can read about Ferrari's method..