Multiplying Simple Rational Expressions

Multiply: $\frac{3x}{4} \times \frac{8}{9x^2}$

Write as a single fraction: $\frac{3x \times 8}{4 \times 9x^2}$ = $\frac{24x}{36x^2}$

Factor out common terms: $\frac{24x}{36x^2} = \frac{24 \cdot x}{36 \cdot x \cdot x}$ = Identify common factors: 12 and $x$

Cancel common factors: $\frac{24}{36} = \frac{2}{3}$ and $\frac{x}{x^2} = \frac{1}{x}$ = $\frac{2}{3x}$

Write final answer: Combine the simplified parts = $\frac{2}{3x}$, where $x \neq 0$

Answer: $\frac{2}{3x}$, where $x \neq 0$

Multiplying with Polynomial Factors

Multiply: $\frac{x^2 - 4}{x + 3} \times \frac{x + 3}{x - 2}$

Factor all expressions completely: $x^2 - 4 = (x+2)(x-2)$ (difference of squares) = $\frac{(x+2)(x-2)}{x+3} \times \frac{x+3}{x-2}$

Write as a single fraction: $\frac{(x+2)(x-2)(x+3)}{(x+3)(x-2)}$ = Numerator and denominator combined

Identify common factors: $(x-2)$ appears in both numerator and denominator $(x+3)$ appears in both numerator and denominator = Two pairs of common factors

Cancel common factors: $\frac{(x+2)\cancel{(x-2)}\cancel{(x+3)}}{\cancel{(x+3)}\cancel{(x-2)}}$ = $x + 2$

State restrictions: Original denominators: $x+3 \neq 0$ and $x-2 \neq 0$ = $x \neq -3$ and $x \neq 2$

Answer: $x + 2$, where $x \neq -3$ and $x \neq 2$

Multiplying Three Rational Expressions

Multiply: $\frac{2x}{x^2 - 1} \times \frac{x + 1}{4} \times \frac{x - 1}{x}$

Factor all expressions: $x^2 - 1 = (x+1)(x-1)$ = $\frac{2x}{(x+1)(x-1)} \times \frac{x+1}{4} \times \frac{x-1}{x}$

Combine into one fraction: $\frac{2x(x+1)(x-1)}{(x+1)(x-1) \cdot 4 \cdot x}$ = $\frac{2x(x+1)(x-1)}{4x(x+1)(x-1)}$

Cancel common factors: $\frac{\cancel{2}\cancel{x}\cancel{(x+1)}\cancel{(x-1)}}{\cancel{4}^2\cancel{x}\cancel{(x+1)}\cancel{(x-1)}}$ = $\frac{1}{2}$

State restrictions: $x \neq 0$, $x \neq 1$, $x \neq -1$ = From all original denominators

Answer: $\frac{1}{2}$, where $x \neq 0$, $x \neq 1$, $x \neq -1$

Mistake: Forgetting to factor before multiplying

Why: Multiplying unfactored polynomials creates larger expressions that are harder to simplify later.

Correct: Always factor each numerator and denominator completely BEFORE multiplying. This makes cancellation much easier.

Mistake: Canceling terms instead of factors

Why: You can only cancel factors (things being multiplied), not terms (things being added).

Correct: $\frac{x+3}{x+5}$ cannot be simplified by canceling $x$. But $\frac{x(x+3)}{x(x+5)} = \frac{x+3}{x+5}$ because $x$ is a factor.

Mistake: Forgetting domain restrictions

Why: Even after cancellation, values that made the original expression undefined are still excluded.

Correct: State all restrictions from the ORIGINAL denominators, not just the simplified form.

Mistake: Adding instead of multiplying

Why: Multiplication and addition of fractions follow different rules.

Correct: For multiplication: multiply across. $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$

Combined Rates in Physics

When calculating compound effects like resistance or velocity changes, multiplying rational expressions models how quantities combine.

If a car travels $\frac{d}{t_1}$ km/h for the first leg and its speed changes by a factor of $\frac{t_1}{t_2}$, the effective rate is $\frac{d}{t_1} \times \frac{t_1}{t_2} = \frac{d}{t_2}$ km/h.

Probability Calculations

Finding the probability of combined independent events requires multiplying rational expressions.

If the probability of event A is $\frac{x}{x+5}$ and event B is $\frac{x+5}{2x}$, then $P(A \text{ and } B) = \frac{x}{x+5} \times \frac{x+5}{2x} = \frac{1}{2}$.

Q: Can I cancel before multiplying?

A: Yes! In fact, it's recommended. You can cancel any factor in any numerator with the same factor in any denominator before multiplying. This keeps numbers smaller and easier to work with.

Q: What if there's nothing to cancel?

A: Just multiply the numerators and denominators as they are. Not every multiplication will have common factors to cancel.

Q: Do I need to find a common denominator?

A: No! Unlike addition and subtraction of fractions, multiplication does not require a common denominator. Just multiply straight across.