Question

Write the polynomial expression in simplest form:
2x-10/x^2 -25 * x^2 +6x +5 /x^2
The simplified form is
(Keep answer in factored form.)



What are the restrictions on the variable?
(Type only the numbers and separate your answers with commas, if needed.)

Answer 1

To simplify the given polynomial expression \(\frac{2x - 10}{x^2 - 25} \times \frac{x^2 + 6x + 5}{x^2}\), we first factor each component where possible.

1. **Factor the first fraction**:
\[
2x - 10 = 2(x - 5)
\]
\[
x^2 - 25 = (x - 5)(x + 5) \quad (\text{difference of squares})
\]
Therefore, the first fraction simplifies to:
\[
\frac{2(x - 5)}{(x - 5)(x + 5)}
\]

2. **Factor the second fraction**:
The expression \(x^2 + 6x + 5\) can be factored as:
\[
x^2 + 6x + 5 = (x + 1)(x + 5)
\]
Thus, the second fraction becomes:
\[
\frac{(x + 1)(x + 5)}{x^2}
\]

3. **Combine the two fractions**:
The overall expression is now:
\[
\frac{2(x - 5)}{(x - 5)(x + 5)} \times \frac{(x + 1)(x + 5)}{x^2}
\]
This can be simplified by multiplying the numerators and denominators:
\[
= \frac{2(x - 5)(x + 1)(x + 5)}{(x - 5)(x + 5)x^2}
\]

4. **Cancel out common factors**:
Here, we can cancel the \((x - 5)\) and \((x + 5)\) from the numerator and the denominator:
\[
= \frac{2(x + 1)}{x^2}
\]

5. **Final expression**:
The simplified polynomial expression is:
\[
\frac{2(x + 1)}{x^2}
\]
Since the question asks for the factored form, we can write it as:
\[
\frac{2(x + 1)}{x \cdot x}
\]
Thus, the simplest form is:
\[
\frac{2(x + 1)}{x^2}
\]

**Restrictions on the variable**:
The expression will have restrictions based on the values that make the denominator zero.

1. From \(x^2\), we have \(x \neq 0\).
2. From the factorization in the first part \((x - 5)(x + 5)\), we have:
- \(x - 5 \neq 0 \Rightarrow x \neq 5\)
- \(x + 5 \neq 0 \Rightarrow x \neq -5\)

Therefore, the restrictions on \(x\) are:
\[
x \neq 0, 5, -5
\]

In numeric format, the restrictions on the variable are:
\[
0, 5, -5
\]