Question

A) Write the polynomial expression in simplest form: 2x−10/x−25×x2+6x+5/x2+10x+21 B) What are the restrictions on the variable? MUST SHOW WORK to earn full credit. (4 points)

Answer 1

To simplify the polynomial expression \( \frac{2x - 10}{x - 25} \times \frac{x^2 + 6x + 5}{x^2 + 10x + 21} \), we will follow these steps:

### A) Simplifying the Expression

1. **Factor each part of the expression:**

- Factor \( 2x - 10 \):
\[
2x - 10 = 2(x - 5)
\]

- Factor \( x^2 + 6x + 5 \):
\[
x^2 + 6x + 5 = (x + 1)(x + 5)
\]
(The factors of 5 that add up to 6 are 1 and 5.)

- Factor \( x^2 + 10x + 21 \):
\[
x^2 + 10x + 21 = (x + 3)(x + 7)
\]
(The factors of 21 that add up to 10 are 3 and 7.)

2. **Rewrite the expression with the factored components:**
\[
\frac{2(x - 5)}{x - 25} \times \frac{(x + 1)(x + 5)}{(x + 3)(x + 7)}
\]

3. **Combine the fractions:**
\[
\frac{2(x - 5)(x + 1)(x + 5)}{(x - 25)(x + 3)(x + 7)}
\]

4. **Notice if any terms can cancel out:**
- In this case, no factors in the numerator cancel with factors in the denominator.

5. **Write the final simplified expression:**
\[
\frac{2(x - 5)(x + 1)(x + 5)}{(x - 25)(x + 3)(x + 7)}
\]

### B) Determining Restrictions on the Variable

Next, we need to identify the restrictions on the variable \( x \) based on the denominators.

1. **Denominator from the first fraction:**
\[
x - 25 \neq 0 \implies x \neq 25
\]

2. **Denominator from the second fraction:**
- \( x + 3 \neq 0 \implies x \neq -3 \)
- \( x + 7 \neq 0 \implies x \neq -7 \)

Thus, the complete restrictions on \( x \) are:
\[
x \neq 25, \quad x \neq -3, \quad x \neq -7
\]

### Final Answers

A) The simplified expression is:
\[
\frac{2(x - 5)(x + 1)(x + 5)}{(x - 25)(x + 3)(x + 7)}
\]

B) The restrictions on the variable are:
\[
x \neq 25, \quad x \neq -3, \quad x \neq -7
\]