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Find all the zeroes of the equation.
x4–6x2–7x–6=0
To find the zeroes of the equation x^4 - 6x^2 - 7x - 6 = 0, we can use factoring.
First, let's factor out any common factors:
x^4 - 6x^2 - 7x - 6 = 0
(x^4 - 7x) - (6x^2 + 6) = 0
x(x^3 - 7) - 6(x^2 + 1) = 0
Now, let's examine each factor separately.
Factor 1:
x = 0
Factor 2:
x^3 - 7 = 0
This is a cubic equation, so we cannot easily factor it further. However, we can see that x = ∛7 satisfies the equation.
Factor 3:
x^2 + 1 = 0
Solving this quadratic equation, we find that there are no real zeroes. However, we can find the complex zeroes by factoring it using the imaginary unit, i:
(x - i)(x + i) = 0
x = i or x = -i
Therefore, the zeroes of the equation x^4 - 6x^2 - 7x - 6 = 0 are:
x = 0, x = ∛7, x = i, x = -i