Chapter 8.6, Problem 5ST

To determine

To find: The polynomial equation.

Answer to Problem 5ST

The equation is $2x^3-x^2-6x+3=0$

Explanation of Solution

Given information:

The integral coefficient is given as:

  $\frac{1}{2},-1\text{ and 3}$

Concept used:

Factor theorem:

The polynomial p( x ) has $r-x$ as a factor if r is a root of equation p( x )

Calculation:

The factor can determine as:

By the factor theorem, the required polynomial must have factors $\begin{array}{l}\left(x-\frac{1}{2}\right),\left[x-\left(-1\right)\right]\text{ and }\left(x-3\right)\\ \frac{1}{2}\end{array}$.

To obtain integers, use $\left(2x-1\right)$ in place of $\left(x-\frac{1}{2}\right)$ , since both produce the root $\frac{1}{2}$

Therefore a solution is,

  $\begin{array}{l}\left(2x-1\right)\left(x+1\right)\left(x-3\right)=0\\ 2x^3-x^2-6x+3=0\end{array}$

Therefore,

The equation is $2x^3-x^2-6x+3=0$