Find one solution graphically and then find the remaining solutions using synthetic division.
-$$2 x ^ { 3 } + 5 x ^ { 2 } - 14 x - 8 = 0$$
A) $4 , - 2,1$
B) $- 4,2 , - 1 / 2$
C) $- 4,2 , - 1$
D) $4 , - 2,1 / 2$

Question

Quizplus · Accepted Answer

To solve the cubic equation $2x^3 + 5x^2 - 14x - 8 = 0$ graphically, we look for points where the graph crosses the x-axis. These points are the roots of the equation. Without the actual graph, we can use the Rational Root Theorem and synthetic division to find the roots. The Rational Root Theorem suggests that possible rational roots are factors of the constant term (-8) divided by factors of the leading coefficient (2), which are $\pm1, \pm2, \pm4, \pm8$ and $\pm1/2, \pm4/2=\pm2$.By trial and error or graphing, we can find that $x = -2$ is a root because substituting $-2$ into the equation gives a result of $0$. Using synthetic division with $x = -2$, we can divide the original polynomial to find the quotient polynomial:$$\begin{array}{r|rrrr}-2 & 2 & 5 & -14 & -8 \\hline& & -4 & 2 & 24 \\hline& 2 & 1 & -12 & 16 \\end{array}$$This gives us a quotient of $2x^2 + x - 12$, which simplifies to $2x^2 + x - 12$. Factoring or using the quadratic formula on this quadratic equation, we find the remaining roots:$$2x^2 + x - 12 = 0$$Solving this quadratic equation, we get:$$x = 2, x = -\frac{1}{2}$$Therefore, the solutions to the original cubic equation are $x = -2, x = 2, x = -\frac{1}{2}$, which matches choice B.

Find One Solution Graphically and Then Find the Remaining Solutions 2x3+5x2−14x−8=02 x ^ { 3 } + 5 x ^ { 2 } - 14 x - 8 = 02x3+5x2−14x−8=0

Find One Solution Graphically and Then Find the Remaining Solutions $2 x ^ { 3 } + 5 x ^ { 2 } - 14 x - 8 = 0$