To determine the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( 3x^3 = 12x \)[/tex], we can follow these steps:
1. Rearrange the equation:
Start with the given equation:
[tex]\[
3x^3 = 12x
\][/tex]
Subtract [tex]\( 12x \)[/tex] from both sides to set the equation to zero:
[tex]\[
3x^3 - 12x = 0
\][/tex]
2. Factor out the common term:
Notice that both terms on the left hand side of the equation have a common factor of [tex]\( 3x \)[/tex]:
[tex]\[
3x(x^2 - 4) = 0
\][/tex]
3. Set each factor to zero:
According to the zero product property, if the product of several factors is zero, at least one of the factors must be zero. So, we set each factor to zero and solve for [tex]\( x \)[/tex]:
[tex]\[
3x = 0 \quad \text{or} \quad x^2 - 4 = 0
\][/tex]
Solving [tex]\( 3x = 0 \)[/tex]:
[tex]\[
x = 0
\][/tex]
Solving [tex]\( x^2 - 4 = 0 \)[/tex]:
[tex]\[
x^2 = 4
\][/tex]
Taking the square root of both sides:
[tex]\[
x = \pm 2
\][/tex]
4. Combine the solutions:
The solutions to the equation are [tex]\( x = 0 \)[/tex], [tex]\( x = 2 \)[/tex], and [tex]\( x = -2 \)[/tex].
Therefore, the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( 3x^3 = 12x \)[/tex] are [tex]\( \boxed{2, -2, 0} \)[/tex], which corresponds to option [tex]\( \text{b} \)[/tex].
