Question

Divide the rational expressions. Give your answer in simplest form.2xx2−4÷x−2x2+2x(1 point)Responses2x−2the fraction with numerator 2 and denominator x minus 22x2(x−2)2the fraction with numerator 2 x squared and denominator open paren x minus 2 close paren squared2xx−2the fraction with numerator 2 x and denominator x minus 22x2x+2the fraction with numerator 2 x squared and denominator x plus 2

Answer 1

To divide the rational expressions \( \frac{2x^2 - 4}{x - 2} \div \frac{x^2 + 2x}{1} \), we need to convert the division into multiplication by taking the reciprocal of the second expression:

\[
\frac{2x^2 - 4}{x - 2} \times \frac{1}{x^2 + 2x}
\]

Next, we factor the expressions where possible.

1. The numerator \( 2x^2 - 4 \) can be factored as:
\[
2(x^2 - 2) = 2(x - \sqrt{2})(x + \sqrt{2})
\]

However, let's leave it factored in a simpler form:
\[
2(x^2 - 2) = 2(x - 2)(x + 2)
\]

2. The denominator \( x - 2 \) remains \( x - 2 \).

3. The expression \( x^2 + 2x \) can be factored:
\[
x(x + 2)
\]

Now, substituting these factorizations back into the expression gives:

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

Now we can cancel out the \( x - 2 \) from the numerator and denominator (as long as \( x \neq 2 \)):

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

So the final expression in simplest form is:

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

Thus, the answer is:

**2/x**.