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. Hint: Use your knowledge of factoring polynomials.

(x2 + 1)(x3 + 2x)(x2 - 64) = 0

Answer 1

`{x}= \pm \: i\: or \: x = 0 \: or \: {x}= \pm √(2)i \: or \: {x}= \pm8`

Explanation:

The given polynomial is

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

By the zero product principle,

`{x}^(2) + 1 = 0 \: or \: {x}^(3) + 2x = 0 \: or \: {x}^(2) - 64 = 0`

Or

`{x}^(2) + 1 = 0 \: or \: x( {x}^(2) + 2)= 0 \: or \: {x}^(2) - 64 = 0`

This implies that;

`{x}^(2)= - 1 \: or \: x = 0 \: or \: ( {x}^(2) + 2)= 0 \: or \: {x}^(2) = 64`

`{x}= \pm √( - 1) \: or \: x = 0 \: or \: {x} = \pm √( - 2) \: or \: {x}= \pm √(64)`

Hence the roots are:

`{x}= \pm \: i\: or \: x = 0 \: or \: {x}= \pm √(2)i \: or \: {x}= \pm8`