Question

factor \(x^4+x^2+25\)

Answer 1

let x^2 = y
y^2+y+25

Answer 2

that doesnt factor nicely

Answer 3

does it have to be nice lol

Answer 4

oh unreal roots

Answer 5

haha no, just want to factor it

Answer 6

wolfram gives
\[x^4+x^2+25=(x^2+3x+5)(x^2-3x+5)\]

Answer 7

which looks nice hm

Answer 8

find its apex. its 0,25

Answer 9

who is apex

Answer 10

the apex of the curve is at 0,25

Answer 11

it has no intercepts so thats annoying, appart from the apex that is

Answer 12

I only can see trying lol x^4 and 25 are perfect square so we wanna find combination that eliminates other turms

Answer 13

(x^2+d1+5)(x^2+d2+5)

Answer 14

thats how i would try but there seems to be some trick to factor this

Answer 15

it looks special to me but idk if the special tricks are any useful because they don't work in general

Answer 16

Hmmm its like a ring

Answer 17

Lets try to factor this
X^4+x^2+9
Seems doesn't factor nice same way

Lets see
X^4+x^2+64
Same

Answer 18

Hmmm so if we have x^4+x2+n^2
It can be factorized nicely as same ur question if 2n_1 is perfect square

Answer 19

I mean 2n minus 1

Answer 20

!_!

Answer 21

(x^2+d1+5)(x^2+d2+5) but since there is no x^3 term, d1=-d2
then looking at the x^2 term by expansion, it is 5 + 5 + d(-d) and equal to 1
so d^2=10-1 and d=3

In general (x^2+dx+n)(x^2-dx+n) will always expend to x^4 + (2n-d^2)*x^2 + n^2
Not so easy to spot but once u see it, u can recognize its symmetry.

Answer 22

Thats what I thought by saying a ring

Answer 23

I think that should be the general method! this is a shortcut im talking about but there is no guarantee that this works always \[\begin{align}x^4+\color{blue}{x^2}+25&= x^4+\color{blue}{10x^2-9x^2}+25\\
&=x^4+10x^2+25 - 9x^2\\
&= (x^2+5)^2-(3x)^2\\
&=(x^2+5+3x)(x^2+5-3x)
\end{align}\]

Answer 24

Nice !!!