Find all the zeroes of the equation. x^4–6x^2–7x–6=0

Answer 1

To find the zeroes of the equation x^4 - 6x^2 - 7x - 6 = 0, we can try factoring the equation, as shown below:

x^4 - 6x^2 - 7x - 6 = (x^4 - 2x^2) - (4x^2 + 7x + 6)
= x^2(x^2 - 2) - (4x^2 + 7x + 6)
= x^2(x^2 - 2) - (4x^2 + 2x + 5x + 6)
= x^2(x^2 - 2) - 2x(2x + 1) - 3(2x + 1)
= (x^2 - 2)(x^2 - 2x - 3)
= (x^2 - 3)(x^2 + x - 2)

Now we have two quadratic equations:
1) x^2 - 3 = 0
2) x^2 + x - 2 = 0

Solving equation 1, we get:
x^2 - 3 = 0
x^2 = 3
x = ±√3

Solving equation 2, we get:
x^2 + x - 2 = 0
(x + 2)(x - 1) = 0
x = -2 or x = 1

Therefore, the zeroes of the equation x^4 - 6x^2 - 7x - 6 = 0 are x = -2, x = 1, x = -√3, and x = √3.