<s>[INST] Interception with $x$-axis - not so trivial? I want to find the interception with the x-axis of the following function:
$f(x) = \frac{1}{4}x^4-x^3+2x$.
So putting $0 = \frac{1}{4}x^4-x^3+2x$ I would get $0 = x(\frac{1}{4}x^3-x^2+2)$ but how to check for the other solutions properly?
 [/INST] $f(x)=\frac14 x^4-x^3+2x=\frac14 x(x^3-4x^2+8)$.
Let $g(x)=x^3-4x^2+8$. Since $g(2)=0$, we know that $x-2$ divides $g(x)$, so $g(x)=(x-2)(x^2+kx-4)$ for some constant $k$. Equating coefficients of $x$ gives $-2k-4=0$ so $k=-2$. Now $g(x)=(x-2)(x^2-2x-4)$.
Let $h(x)=x^2-2x-4=(x-1)^2-5$. Then the zeroes of $h(x)$ are $1+\sqrt 5$ and $1-\sqrt 5$. Hence the zeroes of $f(x)$ are $0$, $2$, $1+\sqrt 5$ and $1-\sqrt 5$.
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