Factorise polynomial (1 Viewer)

[QZP]

How can I factorise this: x^4 + 2x^2 + 4

Answer is (x^2 + root2.x + 2)(x^2 - root2.x + 2)

 

[Trebla]

Hint: write 2x² as 4x² - 2x²

 

[iStudent]

Hint: write 2x² as 4x² - 2x²

This method works (without complex numbers)

Just wondering what made you think of using it (seems very useful)

 

[Chris100]

Wow very nice trebla
Pls tell us the origins of that neat idea

 

[Trebla]

This method works (without complex numbers)

Just wondering what made you think of using it (seems very useful)

To be fair I've seen this question several times before and the answer sorta gives it away (notice it can expand as a difference of two squares)

But yeah the trick is looking ahead and realising that doing such a thing leads to a nice factorisation giving the difference of two squares

 

[dunjaaa]

Trebla's method is like 500 times easier but if you couldn't see it, then here is the solution using complex numbers.

 

[anomalousdecay]

View attachment 29584 Trebla's method is like 500 times easier but if you couldn't see it, then here is the solution using complex numbers.

Its extension 1. OP has to understand Trebla's method, unless OP is doing ext.2

 

[dunjaaa]

his doing ext 2 lol

 

[RealiseNothing]

This method works (without complex numbers)

Just wondering what made you think of using it (seems very useful)

Wow very nice trebla
Pls tell us the origins of that neat idea

It's very similar to the method of completing the square. You can think of it as:

if you let

Now you know that or

So you can use either of these, and it just happens the latter is the easiest to manipulate.

 

[iStudent]

found an alternative method that also works (same thing but i thought it was easier to see)
x^4 + 2x^2 + 4
= x^2(x^2)+2(x^2+2)
= x^2(x^2+2)+2(x^2+2) - 2x^2
= (x^2+2)(x^2+2) - 2x^2
= (x^2+2)^2 - 2x^2
= (x^2-sqrt(2)x+2)(x^2+sqrt(2)+2)

 

[braintic]

It's very similar to the method of completing the square.

It IS completing the square - the original expression has been written as the sum/difference of two terms that can be interpreted as perfect squares - exactly what completing the square involves. Its just that most people are used to seeing only the standard question types.