Find the zeros of the function. State the multiplicity of multiple zeros.
y=4x^3-4x
To find the zeros of the function, we set y equal to zero and solve for x.
0 = 4x^3 - 4x
Factoring out the common factor of 4x, we get:
0 = 4x(x^2 - 1)
Setting each factor equal to zero, we have:
4x = 0 or x^2 - 1 = 0
From 4x = 0, we find x = 0.
From x^2 - 1 = 0, we can factor using the difference of squares:
(x + 1)(x - 1) = 0
Setting each factor equal to zero, we have:
x + 1 = 0 or x - 1 = 0
Solving for x, we find x = -1 or x = 1.
Therefore, the zeros of the function are x = 0, x = -1, and x = 1.
For the multiplicity of the zeros, we can look at the power of each factor in the factored form of the equation.
The factor 4x has a power of 1, meaning it has a multiplicity of 1.
The factor (x + 1) has a power of 1, meaning it also has a multiplicity of 1.
The factor (x - 1) has a power of 1, meaning it also has a multiplicity of 1.
Therefore, all zeros have a multiplicity of 1.