What are the solutions of the equation x⁶ + 6x³ + 5 = 0? Use factoring to solve.

Solution:

Given, the equation is x⁶ + 6x³ + 5 = 0

We have to find the solution of the equation by factoring.

Let x³ = y

Equation becomes y² + 6y + 5 = 0

y² + 5y + y + 5 = 0

y(y + 5) + 1(y + 5) = 0

(y + 5)(y + 1) = 0

To find the solution,

y + 5 = 0

y = -5

y + 1 = 0

y = -1

When, y = -1, 

x³ = (-1)³

Taking cube root,

x = -1

When, y = -5, 

x³ = (-5)³ =

x³ = -125

Taking cube root,

x = ∛-5

Therefore, the solutions are x = -1 and x = ∛-5

What are the solutions of the equation x⁶ + 6x³ + 5 = 0? Use factoring to solve.

Summary:

The solutions to the equation x⁶ + 6x³ + 5 = 0 by factoring are x = -1 and x = ∛-5.