Algebra 2 Flashcards: Operating With Rational Expressions

What this deck covers

This deck focuses on Operating With Rational Expressions, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: What is the result of adding $\frac{a}{b}+\frac{c}{b}$ when denominators match?

Answer: $\frac{a+c}{b}$. Combine numerators over the common denominator.

Flashcard 2: What additional restriction occurs when dividing by $\frac{x-2}{x+1}$?

Answer: $\frac{x-2}{x+1}\neq 0$, so $x\neq 2$ (and also $x\neq -1$). The divisor cannot equal zero for division to be defined.

Flashcard 3: What is $\frac{x-3}{x}\div\frac{x-3}{x+1}$ simplified (as an expression)?

Answer: $\frac{x+1}{x}$. Cancel common factor $(x-3)$ after multiplying by reciprocal.

Flashcard 4: What is $\frac{1}{x}\div\frac{1}{x+5}$ simplified?

Answer: $\frac{x+5}{x}$. Multiply by reciprocal: $\frac{1}{x} \cdot \frac{x+5}{1}$.

Flashcard 5: What is $\frac{x^2-1}{x+1}\div(x-1)$ simplified?

Answer: $1$. Rewrite division as multiplication: $\frac{x^2-1}{x+1} \cdot \frac{1}{x-1}$.

Flashcard 6: What is $\frac{x}{x-1}\div\frac{x+2}{x-1}$ simplified?

Answer: $\frac{x}{x+2}$. Multiply by reciprocal and cancel $(x-1)$.

Flashcard 7: What is $\frac{2}{x}\div\frac{3}{x+1}$ simplified?

Answer: $\frac{2(x+1)}{3x}$. Multiply by reciprocal: $\frac{2}{x} \cdot \frac{x+1}{3}$.

Flashcard 8: What is $\frac{1}{x+1}-\frac{1}{x-1}$ simplified?

Answer: $\frac{-2}{(x+1)(x-1)}$. Use LCD $(x+1)(x-1)$ and subtract: $(x-1-x-1)$.

Flashcard 9: What value must be excluded when simplifying $\frac{(x-3)(x+1)}{x-3}$?

Answer: $x\neq 3$. The factor $(x-3)$ in original expression cannot equal zero.

Flashcard 10: What is the general division rule for $\frac{a}{b}\div\frac{c}{d}$?

Answer: $\frac{a}{b}\cdot\frac{d}{c}$ (with $b\neq 0$, $c\neq 0$, $d\neq 0$). Division means multiplying by the reciprocal.

Flashcard 11: What is $\frac{2}{x+1}-\frac{3}{x-1}$ simplified?

Answer: $\frac{-x-5}{(x+1)(x-1)}$. Use LCD and subtract: $\frac{2(x-1)-3(x+1)}{(x+1)(x-1)}$.

Flashcard 12: What does it mean for rational expressions to be closed under division?

Answer: The quotient is rational if dividing by a nonzero rational expression. Division by zero is excluded, but the result remains rational.

Flashcard 13: What is $\frac{2}{x+1}+\frac{3}{x-1}$ simplified?

Answer: $\frac{5x+1}{(x+1)(x-1)}$. Use LCD and add: $\frac{2(x-1)+3(x+1)}{(x+1)(x-1)}$.

Flashcard 14: What is $\frac{2}{x+1}-\frac{3}{x-1}$ simplified?

Answer: $\frac{-x-5}{(x+1)(x-1)}$. Use LCD and subtract: $\frac{2(x-1)-3(x+1)}{(x+1)(x-1)}$.

Flashcard 15: What is $\frac{1}{x+2}-\frac{1}{x}$ simplified?

Answer: $\frac{-2}{x(x+2)}$. Find LCD $x(x+2)$ and subtract: $\frac{x-(x+2)}{x(x+2)}$.

Flashcard 16: What is $\frac{1}{x^2-1}-\frac{1}{x+1}$ simplified?

Answer: $\frac{-x}{(x-1)(x+1)}$. Factor $x^2-1$ and use LCD: $\frac{1-(x-1)}{(x-1)(x+1)}$.

Flashcard 17: What is $\frac{x^2-9}{x-3}\cdot\frac{1}{x+3}$ simplified?

Answer: $1$. Factor $x^2-9=(x+3)(x-3)$ and cancel both factors.

Flashcard 18: What does it mean for rational expressions to be closed under multiplication?

Answer: The product of rational expressions is rational. The result of multiplying rational expressions is rational.

Flashcard 19: What is $\frac{x}{x^2-1}+\frac{1}{x+1}$ simplified?

Answer: $\frac{2x+1}{(x-1)(x+1)}$. Factor $x^2-1$ and use LCD: $\frac{x+(x-1)}{(x-1)(x+1)}$.

Flashcard 20: What is the rule for dividing by a rational expression $\frac{P(x)}{Q(x)}$?

Answer: Multiply by the reciprocal $\frac{Q(x)}{P(x)}$, with $P(x)\neq 0$. Division by a fraction means multiplying by its reciprocal.

Flashcard 21: What is $\frac{1}{x}-\frac{1}{x+1}$ simplified?

Answer: $\frac{1}{x(x+1)}$. Find LCD $x(x+1)$ and subtract: $\frac{x+1-x}{x(x+1)}$.

Flashcard 22: What is the result of subtracting $\frac{a}{b}-\frac{c}{b}$ when denominators match?

Answer: $\frac{a-c}{b}$. Subtract numerators over the common denominator.

Flashcard 23: What is the general division rule for $\frac{a}{b}\div\frac{c}{d}$?

Answer: $\frac{a}{b}\cdot\frac{d}{c}$ (with $b\neq 0$, $c\neq 0$, $d\neq 0$). Division means multiplying by the reciprocal.

Flashcard 24: What is $\frac{x}{x^2}\cdot\frac{x^3}{x}$ simplified (for $x\neq 0$)?

Answer: $x$. Simplify powers: $\frac{x \cdot x^3}{x^2 \cdot x} = \frac{x^4}{x^3} = x$.

Flashcard 25: What is $\frac{x^2-1}{x+1}\cdot\frac{1}{x-1}$ simplified?

Answer: $1$. Factor $x^2-1=(x+1)(x-1)$ and cancel both factors.

Flashcard 26: What is $\frac{x}{x-1}\cdot\frac{x-1}{x+2}$ simplified?

Answer: $\frac{x}{x+2}$. Cancel the common factor $(x-1)$.

Flashcard 27: What is $\frac{2}{x}\cdot\frac{3}{x+1}$ simplified?

Answer: $\frac{6}{x(x+1)}$. Multiply straight across: $(2\cdot 3)/(x(x+1))$.

Flashcard 28: What is the general multiplication rule for $\frac{a}{b}\cdot\frac{c}{d}$?

Answer: $\frac{ac}{bd}$ (with $b\neq 0$ and $d\neq 0$). Multiply numerators and denominators separately.

Flashcard 29: What is $\frac{x}{x+2}+\frac{2}{x+2}$ simplified?

Answer: $\frac{x+2}{x+2}=1$. Add numerators over common denominator: $(x+2)/(x+2)$.

Flashcard 30: What is $\frac{x}{x^2}+\frac{1}{x}$ simplified (for $x\neq 0$)?

Answer: $\frac{2}{x}$. Rewrite as $\frac{1}{x}+\frac{1}{x}=\frac{2}{x}$.