Question

Solve the equation $x^4-11x^3+44x^2-76x+48=0$, which has equal roots.

Answer 1

Here $f\left(x\right)=x^4-11x^3+44x^2-76x+48$,
$f^{\prime }\left(x\right)=4x^3-33x^3+88x-76$;
and by the ordinary rule we find that the highest common factor of $f\left(x\right)$ and $f^{\prime }\left(x\right)$ is $x-2$; hence $(x-2{)}^2$ is a factor of $f\left(x\right)$; and
$f\left(x\right)=(x-2{)}^2(x^2-7x+12)$
$=(x-2{)}^2(x-3\left)\right(x-4)$;
thus the roots are $2,2,3,4$.