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 restriction must be stated for the rational expression $\frac{P(x)}{Q(x)}$?

Answer: $Q(x)\neq 0$ (exclude values making $Q(x)=0$). The denominator cannot equal zero to avoid division by zero.

Flashcard 2: 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 3: 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 4: 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 5: 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 6: What is $\frac{1}{x}+\frac{1}{x}$ simplified?

Answer: $\frac{2}{x}$. Add fractions with the same denominator by adding numerators.

Flashcard 7: What is $\frac{(x-3)(x+1)}{x-3}$ simplified (as an expression)?

Answer: $x+1$. Cancel the common factor $(x-3)$ from both parts.

Flashcard 8: What is the domain restriction for $\frac{x^2}{(x-1)(x+4)}$?

Answer: $x\neq 1$ and $x\neq -4$. Each factor in the denominator cannot equal zero.

Flashcard 9: What is $\frac{3}{x}-\frac{1}{x}$ simplified?

Answer: $\frac{2}{x}$. Subtract fractions with same denominator: $(3-1)/x$.

Flashcard 10: 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 11: 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 12: 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 13: What is the least common denominator (LCD) of $\frac{1}{x}$ and $\frac{1}{x-2}$?

Answer: $x(x-2)$. Use the product of distinct linear factors.

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

Answer: $\frac{2x-2}{x(x-2)}$. Find common denominator $x(x-2)$ and add numerators.

Flashcard 15: 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 16: 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 17: What is $\frac{(x-2)(x+5)}{(x-2)(x-1)}$ simplified (keep restrictions implied)?

Answer: $\frac{x+5}{x-1}$. Cancel $(x-2)$ from numerator and denominator.

Flashcard 18: 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 19: What is $\frac{x}{x-1}-\frac{1}{x-1}$ simplified?

Answer: $\frac{x-1}{x-1}=1$. Subtract numerators over common denominator: $(x-1)/(x-1)$.

Flashcard 20: What is the domain restriction for $\frac{1}{x-5}$?

Answer: $x\neq 5$. Set $x-5=0$ to find where the denominator is undefined.

Flashcard 21: What operation is valid to simplify $\frac{(x-2)(x+5)}{(x-2)(x-1)}$?

Answer: Cancel the common factor $(x-2)$, with $x\neq 2$. Common factors can be cancelled when not equal to zero.

Flashcard 22: What is $\frac{x}{x^2}-\frac{1}{x}$ simplified (for $x\neq 0$)?

Answer: $0$. Rewrite as $\frac{1}{x}-\frac{1}{x}=0$.

Flashcard 23: What is $\frac{x}{x+1}+\frac{1}{x+1}$ simplified?

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

Flashcard 24: 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 25: What is the LCD of $\frac{1}{x^2-1}$ and $\frac{1}{x-1}$?

Answer: $(x-1)(x+1)$. Since $x^2-1=(x-1)(x+1)$, LCD includes both factors.

Flashcard 26: What is $\frac{2}{x}-\frac{3}{x}$ simplified?

Answer: $\frac{-1}{x}$. Same denominator subtraction: $(2-3)/x = -1/x$.

Flashcard 27: 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 28: What is the first step to simplify a rational expression?

Answer: Factor numerator and denominator completely. Factoring reveals common factors that can be cancelled.

Flashcard 29: What is a rational expression in terms of polynomials $P(x)$ and $Q(x)$?

Answer: $\frac{P(x)}{Q(x)}$ with $Q(x)\neq 0$. A rational expression is the quotient of two polynomials.

Flashcard 30: What is $\frac{5}{x-4}+\frac{1}{x-4}$ simplified?

Answer: $\frac{6}{x-4}$. Add numerators with same denominator: $(5+1)/(x-4)$.

Flashcard 31: What is $\frac{1}{x-1}+\frac{1}{x+1}$ simplified?

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

Flashcard 32: 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 33: 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 34: 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 35: 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 36: 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 37: What does it mean for rational expressions to be closed under addition and subtraction?

Answer: The sum or difference of rational expressions is rational. The result of adding or subtracting is always a rational expression.

Flashcard 38: What is the domain restriction for $\frac{2}{x^2+1}$ over the real numbers?

Answer: No real restrictions (since $x^2+1\neq 0$ for real $x$). $x^2+1>0$ for all real $x$, so no denominator zeros exist.

Flashcard 39: 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 40: What is the LCD of $\frac{1}{(x-1)^2}$ and $\frac{1}{x-1}$?

Answer: $(x-1)^2$. Use the highest power of each factor.

Flashcard 41: 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 42: 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 43: 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 44: 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 45: What is the LCD of $\frac{1}{x^2}$ and $\frac{1}{x^3}$?

Answer: $x^3$. Use the highest power of $x$ among the denominators.

Flashcard 46: What is the LCD of $\frac{1}{x(x+1)}$ and $\frac{1}{x+1}$?

Answer: $x(x+1)$. The LCD is the product of all distinct factors.

Flashcard 47: What is $\frac{2}{x+3}+\frac{1}{x+3}$ simplified?

Answer: $\frac{3}{x+3}$. Add numerators with same denominator: $(2+1)/(x+3)$.

Flashcard 48: 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 49: What is $\frac{x}{x^2}+\frac{1}{x}$ simplified (for $x\neq 0$)?

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

Flashcard 50: 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 51: 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 52: 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 53: What is the domain restriction for $\frac{x+2}{x^2-9}$?

Answer: $x\neq 3$ and $x\neq -3$. Factor $x^2-9=(x-3)(x+3)$ and exclude where factors equal zero.

Flashcard 54: 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.