Question

What are the roots of the polynomial equation x cubed minus 10 x = negative 3 x squared + 24? Use a graphing calculator and a system of equations.
–24, –3, 12
–12, 3, 24
–4, –2, 3
–3, 2, 4

Answer 1

Answer: –4, –2, 3 C.

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Answer 2

Option C: -4, -2, 3 are the roots of the polynomial.

Explanation:

The equation is $x^{3} -10x=-3x^{2} +24$

Now, Adding $3x^{2}$ on both sides, we have,

$x^{3} +3x^{2} -10x=24$

Subtracting 24 from both sides of the equation, we have,

$x^{3} +3x^{2} -10x-24=0$

Solving the equation using synthetic division, we have,

$(x-2)(x^{2} +x-12)=0$

Now, we shall factor $(x^{2} +x-12)$, we have,

$(x+4)(x-3)$

Thus, we have,

$(x+2)(x-3)(x+4)=0$

Solving each factor, we have,

$\begin{aligned} x+2 &=0 \\ x &=-2 \end{aligned}$  and  $\begin{array}{r}{x-3=0} \\{x=3}\end{array}$  and  $\begin{aligned}x+4 &=0 \\x &=-4\end{aligned}$

Thus, the roots are -2,3 and 4.