Question

Fool's problem of the day,

If \(2x^4+x^3-11x^2+x+2=0\), what are the values of \(\large ( x+\frac1x )\) ?

Answer 1

we know x=0 is not a valid solution. therefore:\[\begin{align}
2x^4+x^3-11x^2+x+2&=0\\
(2x^2-5x+2)(x^2+3x+1)&=0\\
\end{align}\]therefore:\[2x^2-5x+2=0\tag{a}\]or:\[x^2+3x+1=0\tag{b}\]
From (a) we get:\[x(2x-5+\frac2x)=0\]\[x(2(x+\frac1x)-5)=0\]therefore, x=0 (reject as this is not valid), or:\[2(x+\frac1x)-5=0\implies x+\frac1x=\frac{5}{2}\]
From (b) we get:\[x(x+3+\frac1x)=0\]\[x(x+\frac1x+3)=0\]therefore, x=0 (reject as this is not valid), or:\[x+\frac1x+3=0\implies x+\frac1x=-3\]

Answer 2

Awesome!

Answer 3

Nice Asnaseer, but I would have just divided the whole thing by \(x^2\)