1. Express the following as a single fraction: 1.a. Given: $$\frac{p - 3}{5} + \frac{p + 4}{6}$$ - Find the common denominator: $$\text{lcm}(5,6) = 30$$ - Rewrite each fraction with denominator 30: $$\frac{p - 3}{5} = \frac{6(p - 3)}{30} = \frac{6p - 18}{30}$$ $$\frac{p + 4}{6} = \frac{5(p + 4)}{30} = \frac{5p + 20}{30}$$ - Add the numerators: $$\frac{6p - 18}{30} + \frac{5p + 20}{30} = \frac{6p - 18 + 5p + 20}{30} = \frac{11p + 2}{30}$$ 1.b. Given: $$\frac{5}{x} - \frac{x + 2}{2x} - \frac{3x + 1}{x + 2}$$ - Find common denominator: $$2x(x + 2)$$ - Rewrite each fraction: $$\frac{5}{x} = \frac{5 \cdot 2(x + 2)}{2x(x + 2)} = \frac{10(x + 2)}{2x(x + 2)} = \frac{10x + 20}{2x(x + 2)}$$ $$\frac{x + 2}{2x} = \frac{(x + 2)(x + 2)}{2x(x + 2)} = \frac{(x + 2)^2}{2x(x + 2)}$$ $$\frac{3x + 1}{x + 2} = \frac{(3x + 1) \cdot 2x}{2x(x + 2)} = \frac{2x(3x + 1)}{2x(x + 2)} = \frac{6x^2 + 2x}{2x(x + 2)}$$ - Combine all: $$\frac{10x + 20}{2x(x + 2)} - \frac{(x + 2)^2}{2x(x + 2)} - \frac{6x^2 + 2x}{2x(x + 2)} = \frac{10x + 20 - (x + 2)^2 - (6x^2 + 2x)}{2x(x + 2)}$$ - Expand $$ (x + 2)^2 = x^2 + 4x + 4 $$: $$10x + 20 - (x^2 + 4x + 4) - 6x^2 - 2x = 10x + 20 - x^2 - 4x - 4 - 6x^2 - 2x$$ - Simplify numerator: $$- x^2 - 6x^2 + 10x - 4x - 2x + 20 - 4 = -7x^2 + 4x + 16$$ - Final expression: $$\frac{-7x^2 + 4x + 16}{2x(x + 2)}$$ 2. Expand and simplify fully: 2.a. Given: $$5x^2 + xy - (x + y)^2$$ - Expand $$ (x + y)^2 = x^2 + 2xy + y^2 $$: $$5x^2 + xy - (x^2 + 2xy + y^2) = 5x^2 + xy - x^2 - 2xy - y^2$$ - Simplify: $$5x^2 - x^2 + xy - 2xy - y^2 = 4x^2 - xy - y^2$$ 2.b. Given: $$\left(\frac{1}{x} - y\right)^3$$ - Use binomial expansion: $$ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 $$ with $$a = \frac{1}{x}, b = y$$ - Calculate each term: $$a^3 = \frac{1}{x^3}$$ $$3a^2b = 3 \cdot \frac{1}{x^2} \cdot y = \frac{3y}{x^2}$$ $$3ab^2 = 3 \cdot \frac{1}{x} \cdot y^2 = \frac{3y^2}{x}$$ $$b^3 = y^3$$ - Combine with signs: $$\frac{1}{x^3} - \frac{3y}{x^2} + \frac{3y^2}{x} - y^3$$