Which of the following expressions simplifies to an equivalent simplified form of 2x^3−2x/x+1 ?(1 point) Responses 2x^3−2x/x−1 Start Fraction 2 x cubed minus 2 x over x minus 1 End Fraction 2x^3+2x^2−4x/x+2 Start Fraction 2 x cubed plus 2 x squared minus 4 x over x plus 2 End Fraction x^2−1/2x^2+2x Start Fraction x squared minus 1 over 2 x squared plus 2 x End Fraction 2x^2+6x/x^2+2x−3

Question

pick one

Answer 1

To simplify the expression \(\frac{2x^3 - 2x}{x + 1}\), we first factor out the numerator:

\[ 2x^3 - 2x = 2x(x^2 - 1) = 2x(x - 1)(x + 1) \]

Thus, we can rewrite our expression:

\[ \frac{2x(x - 1)(x + 1)}{x + 1} \]

Now, we can cancel \((x + 1)\) from the numerator and the denominator (as long as \(x \neq -1\)):

\[ 2x(x - 1) \]

So, the simplified form of \(\frac{2x^3 - 2x}{x + 1}\) is:

\[ 2x(x - 1) = 2x^2 - 2x \]

Now, let's examine the provided options for an equivalent simplified form:

1. \(2x^3 - 2x\) over \(x - 1\) - does not match.
2. \(2x^3 + 2x^2 - 4x\) over \(x + 2\) - does not match.
3. \(x^2 - 1\) over \(2x^2 + 2x\) - does not match.
4. \(2x^2 + 6x\) over \(x^2 + 2x - 3\) - must check this one.

For the last option, we can simplify \(x^2 + 2x - 3\):

\[ x^2 + 2x - 3 = (x + 3)(x - 1) \]

So the expression can be written as:

\[ \frac{2x^2 + 6x}{(x + 3)(x - 1)} \]

However, simplifying \(2x^2 + 6x\) gives:

\[ 2x(x + 3) \]

Therefore, the overall expression looks like:

\[ \frac{2x(x + 3)}{(x + 3)(x - 1)} \]

If \(x + 3\) is not equal to zero (as long as \(x \neq -3\)), we can also cancel:

\[ \frac{2x}{x - 1} \]

None of the options simplify to the same expression as \(\frac{2x^2 - 2x}{1}\).

Thus, to conclude, the correct equivalent simplified expression of \(\frac{2x^3 - 2x}{x + 1}\) is not among the provided choices.