To determine which expressions are equivalent to [tex]\( x^3 - 4x \)[/tex], we will simplify each given expression and compare it to [tex]\( x^3 - 4x \)[/tex].
Expression A: [tex]\( x(x-4) \)[/tex]
[tex]\[
x(x-4) = x^2 - 4x
\][/tex]
This simplifies to [tex]\( x^2 - 4x \)[/tex]. Clearly, [tex]\( x^2 - 4x \)[/tex] is not equivalent to [tex]\( x^3 - 4x \)[/tex].
Expression B: [tex]\( x(x+2)^2 \)[/tex]
[tex]\[
x(x+2)^2 = x(x^2 + 4x + 4)
\][/tex]
[tex]\[
= x \cdot x^2 + x \cdot 4x + x \cdot 4
\][/tex]
[tex]\[
= x^3 + 4x^2 + 4x
\][/tex]
This simplifies to [tex]\( x^3 + 4x^2 + 4x \)[/tex]. Clearly, [tex]\( x^3 + 4x^2 + 4x \)[/tex] is not equivalent to [tex]\( x^3 - 4x \)[/tex].
Expression C: [tex]\( x(x^2-4) \)[/tex]
[tex]\[
x(x^2-4) = x^3 - 4x
\][/tex]
This is already in the required form [tex]\( x^3 - 4x \)[/tex]. Thus, [tex]\( x(x^2-4) \)[/tex] is equivalent to [tex]\( x^3 - 4x \)[/tex].
Expression D: [tex]\( x(x^2-4x) \)[/tex]
[tex]\[
x(x^2-4x) = x^3 - 4x^2
\][/tex]
This simplifies to [tex]\( x^3 - 4x^2 \)[/tex]. Clearly, [tex]\( x^3 - 4x^2 \)[/tex] is not equivalent to [tex]\( x^3 - 4x \)[/tex].
After analyzing each expression, we find that only Expression C: [tex]\( x(x^2-4) \)[/tex] is equivalent to [tex]\( x^3 - 4x \)[/tex].
So, the correct answer is:
- C. [tex]\( x(x^2-4) \)[/tex]
