Answer
The roots of the polynomial are\[\left\{ -1,-2,3-\sqrt{13},3+\sqrt{13} \right\}\]
Work Step by Step
Consider the given polynomial $f\left( x \right)={{x}^{3}}-4{{x}^{2}}-7x+10$.
Determine values of p and q where p are the factors of the constant in the polynomial and q are the factors of the leading coefficient of the polynomial.
$p=\pm 1,\pm 2,\pm 4,\pm 8$
$q=\pm 1$
Calculate $\frac{p}{q}$.
$\frac{p}{q}=\pm 1,\pm 2,\pm 4,\pm 8$
According to the Descartes’s rule of signs, the function $f\left( x \right)={{x}^{3}}-4{{x}^{2}}-7x+10$ has one sign change. Since the function $f\left( x \right)$ has one variations in sign, the function $f\left( x \right)$ has one positive real root.
Further, evaluate $f\left( -x \right):$
$\begin{align}
& f\left( x \right)={{x}^{4}}-3{{x}^{3}}-20{{x}^{2}}-24x-8 \\
& f\left( -x \right)={{\left( -x \right)}^{4}}-3{{\left( -x \right)}^{3}}-20{{\left( -x \right)}^{2}}-24\left( -x \right)-8 \\
& ={{x}^{4}}+3{{x}^{3}}-20{{x}^{2}}+24x-8
\end{align}$
There are two variations in sign. Thus there are 2 negative real zeros or $2-2=0$ negative real zeros.
Test $x=-1$ as a root of the polynomial:
$\begin{align}
& f\left( x \right)={{x}^{4}}-3{{x}^{3}}-20{{x}^{2}}-24x-8 \\
& f\left( -1 \right)={{\left( -1 \right)}^{4}}-3{{\left( -1 \right)}^{3}}-20{{\left( -1 \right)}^{2}}-24\left( -1 \right)-8 \\
& =0
\end{align}$
Divide the equation $f\left( x \right)$ by $\left( x+1 \right)$.
$\frac{{{x}^{4}}-3{{x}^{3}}-20{{x}^{2}}-24x-8}{\left( x+1 \right)}=\left( x+2 \right)\left( {{x}^{2}}-6x-4 \right)$
Thus, the function can be expressed as $f\left( x \right)=\left( x+1 \right)\left( x+2 \right)\left( {{x}^{2}}-6x-4 \right)$.
Equate $f\left( x \right)=\left( x+1 \right)\left( x+2 \right)\left( {{x}^{2}}-6x-4 \right)$ to zero.
$\begin{align}
& \left( x+1 \right)\left( x+2 \right)\left( {{x}^{2}}-6x-4 \right)=0 \\
& \left( x+1 \right)\left( x+2 \right)\left( x-\left( 3-\sqrt{13} \right) \right)\left( x-\left( 3+\sqrt{13} \right) \right)=0 \\
& x=-1,-2,3-\sqrt{13},3+\sqrt{13}
\end{align}$
The solution of the polynomial is $\left\{ -1,-2,3-\sqrt{13},3+\sqrt{13} \right\}$.
