3 votes

Prove that $a=0$ if and only if $b=0$ for the cubic $x^3 + ax^2 + bx + c=0$ whose roots all have the same absolute value.

The problem given is equivalent to the following situation: Given $|p|=|q|=|r|$, prove that $p+q+r=0$ if and only if $pq+qr+rp=0$. ($\rightarrow$) As there are only three variables, the number of ...
Joshua Woo's user avatar
  • 1,183
1 vote

Prove that $a=0$ if and only if $b=0$ for the cubic $x^3 + ax^2 + bx + c=0$ whose roots all have the same absolute value.

I'm proving what @Joshua Woo had written. I think you'll be able to continue from there. $$pq+qr+rp=0$$ then $$2pq+2qr+2rp=0$$ This means that $$(p+q+r)^2=p^2+q^2+r^2+2pq+2qr+2rp=p^2+q^2+r^2$$ Hence $(...
Gwen's user avatar
  • 1,075

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