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Find all the zeroes of the equation.
x^4–6x^2–7x–6=0
To find the zeroes of the equation x^4 – 6x^2 – 7x – 6 = 0, we can use factoring and the quadratic formula.
First, let's rewrite the equation in factored form by grouping terms:
(x^4 – 6x^2) – (7x + 6) = 0
x^2(x^2 – 6) – 1(7x + 6) = 0
x^2(x^2 – 6) – (7x + 6) = 0
Now let's solve each part separately:
1) Solve x^2 = 0:
x^2 = 0
x = 0
So, one zero is x = 0.
2) Solve x^2 – 6 = 0 using the quadratic formula:
x^2 – 6 = 0
a = 1, b = 0, c = -6
x = (-b ± √(b^2 - 4ac)) / (2a)
x = (-0 ± √(0^2 - 4(1)(-6))) / (2(1))
x = (± √(0 + 24)) / 2
x = (± √24) / 2
x = ± √6
So, two more zeroes are x = √6 and x = -√6.
Therefore, the zeroes of the equation x^4 – 6x^2 – 7x – 6 = 0 are:
x = 0, x = √6, x = -√6.