Question

Divide the following rational expression.

y2-4 y+2
y-2 5y-10

Select the correct choice below and fill in the answer box to complete your choice.
(Simplify your answer.)
y2-4 y+2
y-2 5y -10

y2-4
O B.

O A.

y *- 2,-5,10

y+2
y-2 5y-10

y# -2,2

Answer 1

To divide the rational expressions

\[
\frac{y^2 - 4}{y - 2} \div \frac{5y - 10}{1}
\]

we first rewrite the division as multiplication by the reciprocal:

\[
\frac{y^2 - 4}{y - 2} \times \frac{1}{5y - 10}
\]

Next, we can factor both the numerator \(y^2 - 4\) and the denominator \(5y - 10\):

1. \(y^2 - 4\) is a difference of squares:
\[
y^2 - 4 = (y - 2)(y + 2)
\]

2. \(5y - 10\) can be factored out by 5:
\[
5y - 10 = 5(y - 2)
\]

Now we substitute back into the expression:

\[
\frac{(y - 2)(y + 2)}{y - 2} \times \frac{1}{5(y - 2)}
\]

Now we can simplify:

1. Cancel \(y - 2\) in the numerator and denominator (as long as \(y \neq 2\)):
\[
= \frac{y + 2}{5(y - 2)}
\]

Now we have our simplified expression:

\[
\frac{y + 2}{5(y - 2)}
\]

The restrictions on the variable \(y\) are that \(y \neq 2\) (from the original denominator) and that \(y - 2 \neq 0\).

Thus, the final answer is:

\[
\text{Answer: } \frac{y + 2}{5(y - 2)}, \text{ with restrictions } y \neq 2
\]