Option C: -4, -2, 3 are the roots of the polynomial.
Explanation:
The equation is [tex]x^{3} -10x=-3x^{2} +24[/tex]
Now, Adding [tex]3x^{2}[/tex] on both sides, we have,
[tex]x^{3} +3x^{2} -10x=24[/tex]
Subtracting 24 from both sides of the equation, we have,
[tex]x^{3} +3x^{2} -10x-24=0[/tex]
Solving the equation using synthetic division, we have,
[tex](x-2)(x^{2} +x-12)=0[/tex]
Now, we shall factor [tex](x^{2} +x-12)[/tex], we have,
[tex](x+4)(x-3)[/tex]
Thus, we have,
[tex](x+2)(x-3)(x+4)=0[/tex]
Solving each factor, we have,
[tex]$\begin{aligned} x+2 &=0 \\ x &=-2 \end{aligned}$[/tex] and [tex]\begin{array}{r}{x-3=0} \\{x=3}\end{array}[/tex] and [tex]\begin{aligned}x+4 &=0 \\x &=-4\end{aligned}[/tex]
Thus, the roots are -2,3 and 4.
