Question

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

Answer 1

To simplify the expression \( \frac{2x^3 - 2x}{x + 1} \), we can factor out the common terms in the numerator.

1. Factor the numerator: \[ 2x^3 - 2x = 2x(x^2 - 1) = 2x(x - 1)(x + 1) \]

2. Now substitute this back into the fraction: \[ \frac{2x(x - 1)(x + 1)}{x + 1} \]

3. We can cancel \( x + 1 \) from the numerator and denominator (provided \( x \neq -1 \)): \[ 2x(x - 1) \]

4. Therefore, the simplified expression is: \[ 2x^2 - 2x \]

Now we can check which of the provided options simplifies to the same expression \( 2x^2 - 2x \).

1. First option: \( 2x^3 - 2x / (x - 1) \) - This does not simplify to our form.

2. Second option: \( \frac{2x^3 + 2x^2 - 4x}{x + 2} \)

   • The numerator can be factored as \( 2x(x^2 + x - 2) \).
   • However, this does not simplify to match \( 2x^2 - 2x \).

3. Third option: \( \frac{x^2 - 1}{2x^2 + 2x} \)

   • The numerator \( x^2 - 1 \) factors to \( (x - 1)(x + 1) \), and the denominator can be factored to \( 2(x^2 + x) \).
   • It does not match our expression either.

4. Fourth option: \( \frac{2x^2 + 6x}{x^2 + 2x - 3} \)

   • The numerator factors to \( 2x(x + 3) \).
   • The denominator factors to \( (x + 3)(x - 1) \). Cancelling gives \( \frac{2x}{x - 1} \), which is not equivalent.

None of the options provided are equivalent simplified forms of \( 2x^3 - 2x / (x + 1) \).

Thus, the correct answer is none of the given options simplify to the equivalent expression. If one had to choose, it might be wise to compare calculations for any overlooked algebraic mistakes in simplification to confirm the actual equivalences.