Question

follow the instructions for each problem. show all your work and express your answer in simplest form.

1. multiply:

$\frac{x + 3}{x + 5} cdot \frac{2x + 10}{x^2 + 4x + 3}$

1. divide:

$\frac{x - 7}{(x + 2)(x - 2)} div \frac{7 - x}{(x - 2)}$

1. subtract:

$\frac{1}{x - 1} - \frac{2}{x^2 - 1}$

1. simplify:

$\frac{\frac{x - 3}{3x^3}}{\frac{3 - x}{15x^2}}$

Explanation:

Problem 1: Multiply Rational Expressions

Step1: Factor all terms

$\frac{x+3}{x+5} \cdot \frac{2(x+5)}{(x+1)(x+3)}$

Step2: Cancel common factors

Cancel $x+3$ and $x+5$:
$\frac{1}{1} \cdot \frac{2}{(x+1)}$

Step3: Multiply remaining terms

$\frac{2}{x+1}$

Problem 2: Divide Rational Expressions

Step1: Rewrite division as multiplication

Multiply by reciprocal of the second fraction, and factor $-(x-7)=7-x$:
$\frac{x-7}{(x+2)(x-2)} \cdot \frac{x-2}{-(x-7)}$

Step2: Cancel common factors

Cancel $x-7$ and $x-2$:
$\frac{1}{x+2} \cdot \frac{1}{-1}$

Step3: Simplify the product

$-\frac{1}{x+2}$

Problem 3: Subtract Rational Expressions

Step1: Factor denominator

Rewrite $x^2-1$ as $(x-1)(x+1)$:
$\frac{1}{x-1} - \frac{2}{(x-1)(x+1)}$

Step2: Find common denominator

Multiply first term by $\frac{x+1}{x+1}$:
$\frac{x+1}{(x-1)(x+1)} - \frac{2}{(x-1)(x+1)}$

Step3: Subtract numerators

$\frac{(x+1)-2}{(x-1)(x+1)}$

Step4: Simplify numerator

$\frac{x-1}{(x-1)(x+1)}$

Step5: Cancel common factor

Cancel $x-1$:
$\frac{1}{x+1}$

Problem 4: Simplify Complex Fraction

Step1: Rewrite as division

$\frac{x-3}{3x^3} \div \frac{3-x}{15x^2}$

Step2: Rewrite division as multiplication

Multiply by reciprocal, and factor $-(x-3)=3-x$:
$\frac{x-3}{3x^3} \cdot \frac{15x^2}{-(x-3)}$

Step3: Cancel common factors

Cancel $x-3$, $3x^2$:
$\frac{1}{x} \cdot \frac{5}{-1}$

Step4: Simplify the product

$-\frac{5}{x}$

Answer:

1. $\boldsymbol{\frac{2}{x+1}}$
2. $\boldsymbol{-\frac{1}{x+2}}$
3. $\boldsymbol{\frac{1}{x+1}}$
4. $\boldsymbol{-\frac{5}{x}}$