Question

find the zeros of the function. state the multiplicity of multiple zeros.
$y = 2x^3 - 2x$

select the correct choice below and, if necessary, fill in the answer box(es) within your choice.
(use a comma to separate answers as needed.)
a. the numbers \boxed{} are zeros of multiplicity 1 and the numbers \boxed{} are zeros of multiplicity 2.
\bigcirc b. the numbers \boxed{} are zeros of multiplicity 2.
\bigcirc c. the numbers \boxed{} are zeros of multiplicity 1.
\bigcirc d. the numbers \boxed{} are zeros of multiplicity 3.

Explanation:

Step1: Factor out common term

$y=2x(x^2 - 1)$

Step2: Factor difference of squares

$y=2x(x-1)(x+1)$

Step3: Set $y=0$ to find zeros

$2x(x-1)(x+1)=0$

Step4: Solve for each factor

$2x=0 \implies x=0$; $x-1=0 \implies x=1$; $x+1=0 \implies x=-1$

Step5: Identify multiplicities

Each factor has exponent 1, so all zeros have multiplicity 1.

Answer:

C. The numbers $-1, 0, 1$ are zeros of multiplicity 1.