Question

Solve the following equations:
$x^4+8x^3+9x^2-8x-10=0$.

Answer 1

Given equation, $x^4+8x^3+9x^2-8x-10=0$

Consider $f\left(x\right)=x^4+8x^3+9x^2-8x-10$

By inspection we can easily see that $f\left(1\right)=0$ and $f(-1)=0$. Therefore, $(x-1)$ and $(x+1)$ are two factors of the given equation

$\therefore f\left(x\right)=(x-1)(x+1)\cdot g\left(x\right)⟹g\left(x\right)=x^2+8x+10$

We need to find root of $g\left(x\right)=0$

$⟹x^2+8x+10=0⟹x=\frac{-8\pm \sqrt{64-40}}{2}=-4\pm \sqrt{6}$

$\therefore$ the roots of the given equation are $x=1,-1,-4\pm \sqrt{6}$