Question

Solve the following equations which have equal roots:
$x^5-13x^4+67x^3-171x^2+216x-108=0$.

Answer 1

Give equation, $x^5-13x^4+67x^3-171x^2+216x-108=0$

Consider $f\left(x\right)=x^5-13x^4+67x^3-171x^2+216x-108$

$\therefore f^{\prime }\left(x\right)=5x^4-52x^3+201x^2-342x+216$

Now, HCF of $f\left(x\right)$ and $f^{\prime }\left(x\right)$ is $(x-2)$. Hence $2$ is a double root of $f\left(x\right)=0$

$f\left(x\right)$ can be factored as $(x-2{)}^2(x^3-9x^2+27x-27)$

Note that the above 2^nd term is the expansion of $(x-3{)}^3$

$\therefore f\left(x\right)=(x-2{)}^2(x-3{)}^3$

$\therefore$ Roots of the given equation are $2,2,3,3,3$