Find the zeros of the function. State the multiplicity of multiple zeros.
y=12x^3-12x
To find the zeros of the function, we set y equal to zero and solve for x:
0 = 12x^3 - 12x
Factoring out 12x, we get:
0 = 12x(x^2 - 1)
Setting each factor equal to zero, we have:
12x = 0 or x^2 - 1 = 0
Solving the first equation, we find that x = 0. This is a zero of multiplicity 1.
Solving the second equation, we have:
x^2 = 1
Taking the square root of both sides, we get:
x = 1 or x = -1
Both 1 and -1 are zeros of multiplicity 1.
Therefore, the zeros of the function y = 12x^3 - 12x are x = 0 (multiplicity 1), x = 1 (multiplicity 1), and x = -1 (multiplicity 1).
To find the zeros of the function y = 12x^3 - 12x, we set the equation equal to zero:
12x^3 - 12x = 0
We can factor out a common factor of 12x:
12x(x^2 - 1) = 0
Now we have two factors that could equal zero:
1) 12x = 0
Dividing both sides by 12:
x = 0
2) x^2 - 1 = 0
We can solve this quadratic equation by factoring:
(x - 1)(x + 1) = 0
Setting each factor equal to zero:
x - 1 = 0 or x + 1 = 0
Solving for x in each equation:
x = 1 or x = -1
So the zeros of the function y = 12x^3 - 12x are x = 0, x = 1, and x = -1.
To find the multiplicity of multiple zeros, we look at the degree of the factors. The factor x has a multiplicity of 1 since it is to the power of 1, while the factors (x - 1) and (x + 1) each have a multiplicity of 1 as well.