Question

Write the polynomial expression in simplest form 3x+15/x^2 -25 * x^2 -8x +12/ x^2+2x -8

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 polynomial expression \(\frac{3x + 15}{x^2 - 25} \cdot \frac{x^2 - 8x + 12}{x^2 + 2x - 8}\), we start by factoring the polynomials.

1. **Factoring the first fraction: \(\frac{3x + 15}{x^2 - 25}\)**:
- The numerator: \(3x + 15 = 3(x + 5)\)
- The denominator: \(x^2 - 25 = (x - 5)(x + 5)\) (difference of squares)

Thus, the first fraction becomes:
\[
\frac{3(x + 5)}{(x - 5)(x + 5)} = \frac{3}{x - 5} \quad \text{(cancelling } (x + 5) \text{)}
\]

2. **Factoring the second fraction: \(\frac{x^2 - 8x + 12}{x^2 + 2x - 8}\)**:
- The numerator: \(x^2 - 8x + 12 = (x - 6)(x - 2)\) (factors of 12 that sum to -8)
- The denominator: \(x^2 + 2x - 8 = (x + 4)(x - 2)\) (factors of -8 that sum to 2)

Thus, the second fraction becomes:
\[
\frac{(x - 6)(x - 2)}{(x + 4)(x - 2)} = \frac{x - 6}{x + 4} \quad \text{(cancelling } (x - 2) \text{)}
\]

3. **Now, combine the simplified fractions**:
\[
\frac{3}{x - 5} \cdot \frac{x - 6}{x + 4} = \frac{3(x - 6)}{(x - 5)(x + 4)}
\]

The simplified form of the expression is:
\[
\frac{3(x - 6)}{(x - 5)(x + 4)}
\]

4. **Determining the restrictions on the variable**:
The restrictions arise from values that make any denominator zero. This occurs when:
- \(x - 5 = 0 \Rightarrow x = 5\)
- \(x + 4 = 0 \Rightarrow x = -4\)
- \(x - 2 = 0 \Rightarrow x = 2\) (from the second fraction before cancellation)

Thus, the restrictions on the variable are \(5, -4, 2\).

**Final Answers**:
- Simplified form: \(\frac{3(x - 6)}{(x - 5)(x + 4)}\)
- Restrictions: \(5, -4, 2\)