Question

Solve the following equation by identifying all of its roots including all real and complex numbers. In your final answer, include the necessary steps and calculations.

(x^2 + 1)(x^3 + 2x)(x^2 - 64) = 0

Answer 1

This is easily done by factoring the left side further.


`(x^2+1)(x^3+2x)(x^2-64)=(x^2+1)(x^3+2x)(x-8)(x+8)`

`=x(x^2+1)(x^2+2)(x-8)(x+8)`

`=x(x^2+1)(x-i\sqrt2)(x+i\sqrt2)(x-8)(x+8)`

`=x(x-i)(x+i)(x-i\sqrt2)(x+i\sqrt2)(x-8)(x+8)`

Setting equal to zero yields the solutions


`x=0,\pm i,\pm i\sqrt2,\pm8`