Chapter 6.5, Problem 65PPS

To determine

To solve: the given equation.

Answer to Problem 65PPS

  $x=1,-3$ .

Explanation of Solution

Given:

The given equation is $x^6-26x^3-27=0$ .

Calculation:

The given equation is $x^6-26x^3-27=0$ .

In order to solve the equation, factor the equation. Therefore,

  $\begin{array}{l}{x}^6-26x^3-27=0\left\{\text{given equation}\right\}\\ x^6-x^3+27x^3-27=0\left\{\text{express in simplified form}\right\}\\ x^3\left(x^3-1\right)+27\left(x^3-1\right)=0\left\{\text{taking }\left(x^3-1\right)\text{as common}\right\}\\ \left(x^2+27\right)\left(x^3-1\right)=0\left\{\text{use distributive property of product}\right\}\end{array}$

Either $\left(x^3-1\right)=0$ or $\left(x^2+27\right)=0$

when $\left(x^3-1\right)=0$ , then $x^3=1\Rightarrow x=1$

when $\left(x^2+27\right)=0$ , then $x^3=-27\Rightarrow x^3=\sqrt[3]{-27}\Rightarrow x=-3$

Hence, the solution of the given equation is $x=1,-3$ .