Chapter 8.7, Problem 23WE

To determine

To solve: the equation.

Answer to Problem 23WE

The equation have no rational roots.

Explanation of Solution

Given information:

The equation is $\frac{1}{3}{x}^3+\frac{1}{2}{x}^2+\frac{2}{3}x+1=0$ .

Calculation: by using rational root theorem,

The ratio of constant term and coefficient of highest degree term is $\frac{1}{\frac{1}{3}}=3$ .

Thus, the total possible rational roots are $\pm 1,\pm 3$ .

Now use the synthetic division,

$1$	$\frac{1}{3}$	$\frac{1}{2}$	$\frac{2}{3}$	$1$
		$\frac{1}{3}$	$\frac{5}{6}$	$\frac{3}{2}$
	$\frac{1}{3}$	$\frac{5}{6}$	$\frac{3}{2}$	$\frac{5}{2}$

$-1$	$\frac{1}{3}$	$\frac{1}{2}$	$\frac{2}{3}$	$1$
		$-\frac{1}{3}$	$-\frac{1}{6}$	$-\frac{1}{2}$
	$\frac{1}{3}$	$\frac{1}{6}$	$\frac{1}{2}$	$\frac{1}{2}$

$3$	$\frac{1}{3}$	$\frac{1}{2}$	$\frac{2}{3}$	$1$
		$1$	$\frac{9}{2}$	$\frac{31}{2}$
	$\frac{1}{3}$	$\frac{3}{2}$	$\frac{31}{6}$	$\frac{33}{2}$

$-3$	$\frac{1}{3}$	$\frac{1}{2}$	$\frac{2}{3}$	$1$
		$-1$	$\frac{3}{2}$	$-\frac{13}{2}$
	$\frac{1}{3}$	$-\frac{1}{2}$	$\frac{13}{6}$	$-\frac{11}{2}$

None of these rational roots satisfy the equation.

Thus, the equation have no rational roots.