1. **Problem:** Simplify each expression and determine any restrictions on the variables. --- ### a) Simplify \( \frac{3x^2}{yx} \) 1. Write the expression: \( \frac{3x^2}{yx} \) 2. Factor and cancel common terms: \( \frac{3\cancel{x}x}{y\cancel{x}} = \frac{3x}{y} \) 3. **Restrictions:** The denominator cannot be zero, so \( y \neq 0 \) and \( x \neq 0 \). --- ### b) Simplify \( \frac{x - 3}{x^2 + 3x - 18} \) 1. Factor the denominator: \( x^2 + 3x - 18 = (x + 6)(x - 3) \) 2. Write the expression: \( \frac{x - 3}{(x + 6)(x - 3)} \) 3. Cancel common factor \( x - 3 \): \( \frac{\cancel{x - 3}}{(x + 6)\cancel{(x - 3)}} = \frac{1}{x + 6} \) 4. **Restrictions:** Denominator factors cannot be zero, so \( x \neq -6 \) and \( x \neq 3 \). --- ### c) Simplify \( \frac{x^2 + 10x + 21}{x + 3} \) 1. Factor numerator: \( x^2 + 10x + 21 = (x + 7)(x + 3) \) 2. Write the expression: \( \frac{(x + 7)(x + 3)}{x + 3} \) 3. Cancel common factor \( x + 3 \): \( \frac{(x + 7)\cancel{(x + 3)}}{\cancel{x + 3}} = x + 7 \) 4. **Restrictions:** Denominator cannot be zero, so \( x \neq -3 \). --- ### d) Simplify \( \frac{x + 1}{x^2 + 3x + 2} \) 1. Factor denominator: \( x^2 + 3x + 2 = (x + 1)(x + 2) \) 2. Write the expression: \( \frac{x + 1}{(x + 1)(x + 2)} \) 3. Cancel common factor \( x + 1 \): \( \frac{\cancel{x + 1}}{\cancel{x + 1}(x + 2)} = \frac{1}{x + 2} \) 4. **Restrictions:** Denominator factors cannot be zero, so \( x \neq -1 \) and \( x \neq -2 \). --- **Final simplified expressions and restrictions:** - a) \( \frac{3x}{y} \), with \( x \neq 0, y \neq 0 \) - b) \( \frac{1}{x + 6} \), with \( x \neq -6, 3 \) - c) \( x + 7 \), with \( x \neq -3 \) - d) \( \frac{1}{x + 2} \), with \( x \neq -1, -2 \)