Chapter 8.7, Problem 16OE

To determine

To state: that equation have no rational roots.

Answer to Problem 16OE

None of the possible roots satisfy the equation so the equation have no rational roots.

Explanation of Solution

Given information:

The equation is $x^3+3x-3=0$ .

the possible rational roots of equation can be given by:

  $\begin{array}{l}\text{rational roots}=\frac{\text{constant term}}{\text{coffiecient of highest degree }}\\ \text{rational roots}=\frac{3}{1}\end{array}$

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

Now, use synthetic division,

$1$	$1$	$0$	$3$	$-3$
		$1$	$1$	$4$
	$1$	$1$	$4$	$1$

$-1$	$1$	$0$	$3$	$-3$
		$-1$	$1$	$-4$
	$1$	$-1$	$4$	$-7$

$3$	$1$	$0$	$3$	$-3$
		$3$	$9$	$36$
	$1$	$3$	$12$	$33$
$-3$	$1$	$0$	$3$	$-3$
		$-3$	$9$	$-36$
	$1$	$-3$	$12$	$-39$

Thus, none of the possible rational roots satisfy the equation.

Thus, the equation have no rational roots.