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STEP Support Programme

Assignment 2 - | Warm up part (iv)

Hello,

I have worked through the warm up question in assignment 2 but am slightly unsure about part (iv).

I took the difference of squares and got x^4 + x^2 - 2x + 1. However I am not sure how to use this to "hence" derive the solutions for x^4 + 1 = 0.

Any guidance is appreciated!

If you have simplified $(x^2-\sqrt{2}x+1)(x^2+\sqrt{2}x+1)$ to $x^4+x^2-2x+1$, I think you might need to go over your working again.

Apologies, I get the following:

$$(x^2-\sqrt{2x}+1)(x^2+\sqrt{2x}+1) = x^4+2x^2-2x+1$$

Just unsure how this result can be used to get to the solutions. Thanks!

SPOILERS!!!
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The easiest way (IMO) to simplify this is to use difference of two squares:
\[
(x^2-\sqrt{2} x + 1)(x^2+\sqrt{2} x + 1)=\\
(x^2 + 1 -\sqrt{2} x)(x^2+ 1 +\sqrt{2} x)=\\
(x^2+1)^2 - (\sqrt{2}x)^2=\\
x^4 + 2x^2 + 1 - 2x^2
\]

Thank you!

I misread $$\sqrt{2}x$$ as $$\sqrt{2x}$$ (frustrating!)