Chapter 7.4, Problem 27WE

To determine

Write a fourth-degree equation that has roots $\pm \frac{2}{3}\text{ and }\pm \frac{1}{4}$ .

Answer to Problem 27WE

  $144x^4-73x^2+4=0$

Explanation of Solution

Given:

The fourth-degree equation $4x^4-73x^2+144=0$ has roots $\pm \frac{3}{2}\text{ and }\pm 4$ .

Calculation:

Replace $x\text{ by }{x}^{-1}$ in the equation $4x^4-73x^2+144=0$ and the roots of the resulting equation will be multiplicative inverse of the roots of the original equation.

The roots of the original equation are $\pm \frac{3}{2}\text{ and }\pm 4$ .

So, the roots of the equation obtained by replacing $x\text{ by }{x}^{-1}$ are $\pm \frac{1}{\left(\frac{3}{2}\right)}\text{=}\pm \frac{2}{3}\text{ and }\pm \frac{1}{4}$

  $\begin{array}{l}4x^4-73x^2+144=0\\ x\to x^{-1}\\ \Rightarrow 4{\left(x^{-1}\right)}^4-73{\left(x^{-1}\right)}^2+144=0\\ \Rightarrow 4x^{-4}-73x^{-2}+144=0\\ \Rightarrow \frac{4}{x^4}-\frac{73}{x^2}+144=0\\ \Rightarrow 144x^4-73x^2+4=\mathrm{0...............}[\text{multiply each side by }{x}^4]\end{array}$

So, the roots of the equation $144x^4-73x^2+4=0$ are $\pm \frac{2}{3}\text{ and }\pm \frac{1}{4}$ .