4

We have the function $x^3 + px + q = 0$, where $p$ and $q$ are known real numbers and $x$ is an unknown real number.

Put $x = u + v$ and write it out. If $3uv+p=0$, can you find another equation for $u$ and $v$?

So, for the first step: $u^3 + 3u^2v + 3uv^2 + v^3 + px + q$. But I don't know what to do for the second step. In all honesty, I don't really understand what is meant by I assume the equation has to be simplified.

How would I do it? A long division with $3uv+p$ or something?

Community
  • 1
Preet Singh
  • 73

1 Answers1

4

You can also substitute $x = u+v$ into $px$. Also, $p = -3uv$ from the equation $3uv + p = 0$: $$u^3 + 3u^2v + 3uv^2 + v^3 + \underbrace{-3uv}_p\,(\underbrace{u + v}_x) + q = 0$$

Since $q$ is known, the above is an equation in $u$ and $v$, but can be simplified to $$(u^3 + v^3) = -q$$

amWhy
  • 209,954