What are the solutions to the equation x^6 + 6x^3 + 5 = 0, and how can we solve it using factoring?

by

To solve the equation x6 + 6x3 + 5 = 0 using factoring, we can start by recognizing that this polynomial can be simplified through substitution.

Let y = x3. Then, the equation transforms into:

y2 + 6y + 5 = 0

This is now a quadratic equation in terms of y. We can factor this quadratic:

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

Setting each factor equal to zero gives us:

  1. y + 1 = 0 → y = -1
  2. y + 5 = 0 → y = -5

Now we substitute back for y

  1. x3 = -1 → x = -1
  2. x3 = -5 → x = –
    oot{3}{5}

Therefore, the solutions to the original equation x6 + 6x3 + 5 = 0 are:

  • x = -1
  • x = –
    oot{3}{5}

In conclusion, we successfully factored the polynomial after a substitution, allowing us to find the roots of the equation efficiently!

Leave a Comment