Operating with Rational Expressions - Algebra 1

1. What restriction must be stated for a rational expression $\frac{p(x)}{q(x)}$?

   $q(x)\ne 0$ (exclude values that make the denominator $0$). The denominator cannot equal zero to avoid undefined expressions.

2. What is $\frac{1}{x+1}-\frac{1}{x-1}$ written as a single fraction?

   $\frac{-2}{x^2-1}$. Use LCD $x^2-1$ to get $\frac{x-1-(x+1)}{x^2-1}$.

3. What is the quotient $\frac{2}{x}\div\frac{3}{x+1}$ simplified?

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

4. What is $\frac{2}{x}-\frac{1}{2x}$ written as a single fraction?

   $\frac{3}{2x}$. Use LCD $2x$: $\frac{4}{2x}-\frac{1}{2x}=\frac{3}{2x}$.

5. What is $\frac{1}{x+1}+\frac{1}{x-1}$ written as a single fraction?

   $\frac{2x}{x^2-1}$. Use LCD $ (x+1)(x-1)=$x^2$-1 $ to get $\frac{x-1+x+1}{x^2-1}$

6. What is $\frac{2}{x-3}-\frac{x}{x-3}$ simplified?

   $\frac{2-x}{x-3}$. Subtract numerators: $\frac{2-x}{x-3}$.

7. What is $\frac{x-3}{x^2-9}$ simplified?

   $\frac{1}{x+3}$ (with $x\ne 3$ and $x\ne -3$). Factor $x^2-9=(x-3)(x+1)$ and cancel $(x-3)$.

8. What does it mean for rational expressions to be closed under multiplication?

   The product of two rational expressions is a rational expression. Multiplying rational expressions produces another rational expression.

9. What is $\frac{x}{x+1}+\frac{1}{x+1}$ simplified?

   $\frac{x+1}{x+1}=1$ (with $x\ne -1$). Adding the fractions gives $\frac{x+1}{x+1}$ which simplifies to $1$.

10. What is $\frac{5}{x}-\frac{2}{x}$ simplified?

    $\frac{3}{x}$. Subtract numerators when denominators are the same.

11. What is the least common denominator (LCD) of $\frac{1}{x}$ and $\frac{1}{x-2}$?

    $x(x-2)$. Take the product of distinct denominators for the LCD.

12. What is $\frac{(x-1)(x+4)}{x-1}$ simplified?

    $x+4$ (with $x \neq 1$). Cancel the common factor $(x-1)$.

13. What is a rational expression in terms of polynomials $p(x)$ and $q(x)$?

    $\frac{p(x)}{q(x)}$ where $p(x),q(x)$ are polynomials and $q(x)\ne 0$. This is the standard form where polynomials create a fraction.

14. What is $\frac{2x}{x^2}$ simplified?

    $\frac{2}{x}$ (with $x\ne 0$). Cancel one factor of $x$ from numerator and denominator.

15. What is $\frac{1}{x-3}-\frac{1}{x+3}$ written as a single fraction?

    $\frac{6}{x^2-9}$. Use LCD $x^2-9$ to get $\frac{x+3-(x-3)}{x^2-9}=\frac{6}{x^2-9}$.

16. What is the rule for dividing rational expressions $\frac{a}{b}\div\frac{c}{d}$?

    $\frac{a}{b}\cdot\frac{d}{c}$ with $b\ne 0$ and $c\ne 0$ and $d\ne 0$. Division becomes multiplication by the reciprocal of the divisor.

17. What is $\frac{1}{x}+\frac{2}{3x}$ written as a single fraction?

    $\frac{5}{3x}$. Use LCD $3x$: $\frac{3}{3x}+\frac{2}{3x}=\frac{5}{3x}$.

18. What is $\frac{a}{b}\cdot\frac{c}{d}$ simplified in one step?

    $\frac{ac}{bd}$ (with $b\ne 0$ and $d\ne 0$). Multiply numerators together and denominators together.

19. What is $\frac{2}{x}+\frac{3}{x}$ simplified?

    $\frac{5}{x}$. Add numerators when denominators are the same.

20. What condition is required for closure under division of rational expressions?

    Division is allowed only by a nonzero rational expression. Cannot divide by zero, so the divisor must be nonzero.

21. What is $\frac{1}{x}+\frac{1}{x-1}$ written as a single fraction?

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

22. What does it mean for rational expressions to be closed under addition and subtraction?

    The sum or difference of two rational expressions is a rational expression. Adding or subtracting rational expressions yields another rational expression.

23. What is $\frac{x^2-1}{x+1}$ simplified?

    $x-1$ (with $x\ne -1$). Factor $x^2-1=(x-1)(x+1)$ and cancel $(x+1)$.

24. What is $\frac{x^2-1}{x+1}\div(x-1)$ simplified?

    $1$ (with $x\neq -1$ and $x\neq 1$). Rewrite division as multiplication by reciprocal, then simplify.

25. What is $\frac{4}{x^2-9}+\frac{1}{x+3}$ written as a single fraction?

    $\frac{x+7}{x^2-9}$. Use LCD $x^2-9=(x+3)(x-3)$ to combine the fractions.

26. What is $\frac{x}{x+2}\cdot\frac{x+2}{3}$ simplified?

    $\frac{x}{3}$ (with $x\ne -2$). The $(x+2)$ terms cancel, leaving $\frac{x}{3}$.

27. What is the domain restriction for $\frac{1}{x-3}$?

    $x\ne 3$. Setting the denominator equal to zero: $x-3=0$ gives $x=3$.

28. What is $\frac{x^2+5x}{x}$ simplified?

    $x+5$ (with $x\ne 0$). Factor out $x$ from numerator: $\frac{x(x+5)}{x}$.

29. Identify the missing condition for $\frac{a}{b}\div\frac{c}{d}$ to be defined.

    $\frac{c}{d}\ne 0$ (equivalently $c\ne 0$ and $d\ne 0$). The divisor must be nonzero for division to be defined.

30. What is $\frac{x+2}{x}-\frac{2}{x}$ simplified?

    $\frac{x}{x}=1$ (with $x\ne 0$). Subtracting gives $\frac{x+2-2}{x}=\frac{x}{x}=1$.

31. What is the product $\frac{2}{x}\cdot\frac{3}{x+1}$ simplified?

    $\frac{6}{x(x+1)}$. Multiply numerators and denominators directly.

32. What is the LCD of $\frac{1}{x^2}$ and $\frac{1}{x}$?

    $x^2$. Use the highest power of $x$ that appears in either denominator.

33. What is the reciprocal of the rational expression $\frac{a}{b}$ (with $a\ne 0$ and $b\ne 0$)?

    $\frac{b}{a}$. Flip the numerator and denominator to get the reciprocal.

34. What is $\frac{x^2-4}{x-2}$ simplified?

    $x+2$ (with $x \neq 2$). Factor $x^2-4=(x-2)(x+2)$ and cancel $(x-2)$.

35. What is $\frac{x}{x^2-1}+\frac{1}{x-1}$ written as a single fraction?

    $\frac{2x+1}{x^2-1}$. Rewrite $\frac{1}{x-1}$ with LCD $x^2-1$ then add.

36. What is $\frac{x^2-9}{x-3}$ simplified?

    $x+3$ (with $x\ne 3$). Factor $x^2-9=(x-3)(x+1)$ and cancel $(x-3)$.

37. What is $\frac{a}{b}+\frac{c}{b}$ simplified, assuming $b\ne 0$?

    $\frac{a+c}{b}$. Combine fractions with the same denominator.

38. What is $\frac{x^2-9}{x}\cdot\frac{x}{x-3}$ simplified?

    $x+3$ (with $x\ne 0$ and $x\ne 3$). Factor $x^2-9=(x-3)(x+1)$, then cancel common factors.

39. What is $\frac{a}{b}-\frac{c}{b}$ simplified, assuming $b\ne 0$?

    $\frac{a-c}{b}$. Combine fractions with the same denominator.

40. What is $\frac{3x^2y}{6xy^2}$ simplified?

    $\frac{x}{2y}$ (with $x\ne 0$ and $y\ne 0$). Cancel common factors: $3x$ and $y$.

41. What is $\frac{1}{x}\div\frac{1}{x-3}$ simplified?

    $\frac{x-3}{x}$ (with $x\ne 0$ and $x\ne 3$). Multiply by reciprocal: $\frac{1}{x}\cdot\frac{x-3}{1}$.

42. What is $\frac{x}{x^2-1}-\frac{1}{x+1}$ written as a single fraction?

    $\frac{-1}{x^2-1}$. Rewrite $\frac{1}{x+1}$ with LCD $x^2-1$ then subtract.

43. What is $\frac{x^2-4}{x}\div\frac{x-2}{x+2}$ simplified?

    $\frac{(x+2)^2}{x}$ (with $x \neq 0, 2, -2$). Multiply by reciprocal and simplify by canceling and factoring.

44. What is $\frac{a}{b}\div\frac{c}{d}$ rewritten using multiplication?

    $\frac{a}{b}\cdot\frac{d}{c}$ (require $c\ne 0$). Division by a fraction equals multiplication by its reciprocal.

45. What is $\frac{1}{x}-\frac{1}{x-1}$ written as a single fraction?

    $\frac{-1}{x(x-1)}$. Use LCD $x(x-1)$ and combine: $\frac{x-1-x}{x(x-1)}$.

46. What is the LCD of $\frac{1}{(x-1)^2}$ and $\frac{1}{x-1}$?

    $(x-1)^2$. Use the highest power of $(x-1)$ that appears.

47. What is the LCD of $\frac{1}{x^2-1}$ and $\frac{1}{x-1}$?

    $x^2-1$ (since $x^2-1=(x-1)(x+1)$). Since $x^2-1$ contains $(x-1)$ as a factor, it's the LCD.

48. What is $\frac{x}{x-2}+\frac{2}{x-2}$ simplified?

    $\frac{x+2}{x-2}$. Add numerators since denominators are the same.

49. What is $\frac{x}{5}\div\frac{x}{2}$ simplified (assume $x\ne 0$)?

    $\frac{2}{5}$. Multiply by reciprocal and cancel: $\frac{x}{5}\cdot\frac{2}{x}$.

50. What is $\frac{3}{x+2}+\frac{1}{x+2}$ simplified?

    $\frac{4}{x+2}$. Add numerators since denominators are identical.

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