Question

Write the polynomial expression in simplest form:



\[ \frac{x^2 - 1}{x + 1} \cdot \frac{x + 5}{5x - 5} \]



A. \( x + 1 \)

B. \( x + 5 \)

C. \( x \)

D. \( \frac{x + 5}{5} \)

Answer 1

To simplify the given polynomial expression

\[
\frac{x^2 - 1}{x + 1} \cdot \frac{x + 5}{5x - 5},
\]

we start by factoring the components of the expression:

1. **Factor \( x^2 - 1 \):**
\[
x^2 - 1 = (x - 1)(x + 1) \quad \text{(difference of squares)}
\]

2. **Factor \( 5x - 5 \):**
\[
5x - 5 = 5(x - 1)
\]

Now we can substitute the factored terms back into the expression:
\[
\frac{(x - 1)(x + 1)}{x + 1} \cdot \frac{x + 5}{5(x - 1)}.
\]

Next, we can simplify:

- The \( x + 1 \) in the numerator and the denominator cancels out:
\[
\frac{(x - 1)\cancel{(x + 1)}}{\cancel{(x + 1)}} \cdot \frac{x + 5}{5(x - 1)} = \frac{x - 1}{1} \cdot \frac{x + 5}{5(x - 1)}.
\]

- The \( x - 1 \) in the numerator of the first fraction and the denominator of the second fraction also cancels out:
\[
\frac{\cancel{(x - 1)}}{1} \cdot \frac{x + 5}{5\cancel{(x - 1)}} = \frac{x + 5}{5}.
\]

Thus, the expression simplifies to:
\[
\frac{x + 5}{5}.
\]

The final answer is:
\[
\boxed{\frac{x + 5}{5}}.
\]