Rewriting Rational Expressions - Algebra 1

1. What is $\frac{x^2+1}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

   $x+\frac{1}{x}$. Split each term: $\frac{x^2}{x}+\frac{1}{x}=x+\frac{1}{x}$.

2. What is $\frac{x^2-1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

   $x-1$. Factor as $ (x+1)(x-1) $ then cancel $ (x+1) $.

3. What is $\frac{x^3-1}{x^2+x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

   $x-1$. Factor $x^3-1=(x-1)(x^2+x+1)$ then cancel denominator.

4. What is $\frac{x^3+x^2-x-1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

   $x^2-1$. Factor and divide: $(x+1)(x^2-1)\div(x+1)=x^2-1$.

5. What is $\frac{x^3+2x+1}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

   $x+\frac{2x+1}{x^2}$. Split first term then combine remainder: $x+\frac{2x+1}{x^2}$.

6. What is $\frac{x^3-3x+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

   $x^2-3+\frac{2}{x}$. Split each term: $\frac{x^3}{x}-\frac{3x}{x}+\frac{2}{x}$.

7. What is $\frac{x^3+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

   $x^2+\frac{2}{x}$. Split each term: $\frac{x^3}{x}+\frac{2}{x}=x^2+\frac{2}{x}$.

8. What is $\frac{x^3-x^2+x-1}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

   $x^2+1$. Factor and divide: $(x-1)(x^2+1)\div(x-1)=x^2+1$.

9. What is $\frac{x^3+2x^2-5x+6}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

   $x^2+2x-5+\frac{6}{x}$. Split each term: $x^2+2x-5+\frac{6}{x}$.

10. What is $\frac{x^2+4x+7}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+2+\frac{3}{x+2}$. Long division gives quotient $x+2$ with remainder $3$.

11. What is $\frac{x^3+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x^2-x+1$. Factor $x^3+1=(x+1)(x^2-x+1)$ then cancel $(x+1)$.

12. What is $\frac{x^2-4x+1}{x-2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x-2-\frac{3}{x-2}$. Long division gives quotient $x-2$ with remainder $-3$.

13. Identify the missing term form needed to divide $2x^4-x^2+5$ by $x^2+1$ using long division.

    Rewrite as $2x^4+0x^3-x^2+0x+5$. Add zero coefficients for missing $x^3$ and $x$ terms.

14. What is $\frac{x^3+2x^2+3}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+2+\frac{3}{x^2}$. Split each term: $\frac{x^3}{x^2}+\frac{2x^2}{x^2}+\frac{3}{x^2}$.

15. What is $\frac{x^3+1}{x^2-x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+1$. Factor $x^3+1=(x+1)(x^2-x+1)$ then cancel denominator.

16. What is $\frac{2x^3+3x^2+x+5}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $2x+3+\frac{-x+2}{x^2+1}$. Long division gives quotient $2x+3$ with remainder $-x+2$.

17. What is $\frac{3x^2-2x+4}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $3x+1+\frac{5}{x-1}$. Long division gives quotient $3x+1$ with remainder $5$.

18. What is the remainder degree condition when dividing by a linear $b(x)$ with $\deg(b(x))=1$?

    $r(x)$ is a constant (degree $0$). Linear divisor means remainder has degree less than 1.

19. What is $\frac{4x^2+1}{2x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $2x+\frac{1}{2x}$. Split each term: $\frac{4x^2}{2x}+\frac{1}{2x}=2x+\frac{1}{2x}$.

20. What is $\frac{x^2+3x+2}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+3+\frac{2}{x}$. Split each term: $\frac{x^2}{x}+\frac{3x}{x}+\frac{2}{x}$.

21. What is $\frac{x^2+5x+6}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+3$. Factor as $(x+2)(x+3)$ then cancel $(x+2)$.

22. What is the remainder degree condition when dividing by a quadratic $b(x)$ with $\deg(b(x))=2$?

    $r(x)$ has form $ax+b$. Quadratic divisor means remainder has degree less than 2.

23. What is $\frac{x^3+2x}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x^2+2$. Split each term: $\frac{x^3}{x}+\frac{2x}{x}=x^2+2$.

24. What is the name of the process used to rewrite $\frac{a(x)}{b(x)}$ as $q(x)+\frac{r(x)}{b(x)}$?

    Polynomial long division. The systematic method for dividing polynomials with remainder.

25. What is $\frac{x^3-2x^2+4}{x-2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x^2+\frac{4}{x-2}$. Long division gives quotient $x^2$ with remainder $4$.

26. What is the rewritten form of $\frac{a(x)}{b(x)}$ after division with remainder?

    $\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$. Standard quotient-plus-remainder form after polynomial division.

27. What is $\frac{x^2-1}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+1$. Factor as $(x-1)(x+1)$ then cancel $(x-1)$.

28. What is $\frac{x^2+5x+5}{x+2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+3-\frac{1}{x+2}$. Long division gives quotient $x+3$ with remainder $-1$.

29. What is $\frac{x^2-4}{x-2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+2$. Factor as $(x-2)(x+2)$ then cancel $(x-2)$.

30. What is $\frac{x^4+1}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x^2+\frac{1}{x^2}$. Split each term: $\frac{x^4}{x^2}+\frac{1}{x^2}=x^2+\frac{1}{x^2}$.

31. What is $\frac{3x^2+6}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $3x+\frac{6}{x}$. Split each term: $\frac{3x^2}{x}+\frac{6}{x}=3x+\frac{6}{x}$.

32. What is $\frac{x^4-x}{x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x^3-1$. Factor out $x$: $\frac{x(x^3-1)}{x}=x^3-1$.

33. What is $\frac{x^3+4x^2}{x^2}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+4$. Split each term: $\frac{x^3}{x^2}+\frac{4x^2}{x^2}=x+4$.

34. What is $\frac{x^2+2x+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+1$. Factor as $(x+1)^2$ then cancel $(x+1)$.

35. Identify the missing term form needed to divide $x^3+2x+1$ by $x+1$ using long division.

    Rewrite as $x^3+0x^2+2x+1$. Add zero coefficient for missing $x^2$ term.

36. What is $\frac{x^3+x^2+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+\frac{x^2-x+1}{x^2+1}$. Long division gives quotient $x$ with remainder $x^2-x+1$.

37. What is $\frac{x^3+3x^2+3x+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x^2+2x+1$. Recognize as $(x+1)^3$ expanded then divide by $(x+1)$.

38. What is $\frac{2x^2+3x+4}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $2x+1+\frac{3}{x+1}$. Long division gives quotient $2x+1$ with remainder $3$.

39. What is the required relationship between degrees in $\frac{a(x)}{b(x)}=q(x)+\frac{r(x)}{b(x)}$?

    $\deg(r(x))<\deg(b(x))$. The remainder degree must be less than the divisor degree.

40. What is $q(x)$ called in $a(x)=b(x)q(x)+r(x)$?

    The quotient polynomial $q(x)$. The result when $a(x)$ is divided by $b(x)$.

41. What is $\frac{x^3+2x^2+x+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+2+\frac{x-1}{x^2+1}$. Long division gives quotient $x+2$ with remainder $x-1$.

42. What is $r(x)$ called in $a(x)=b(x)q(x)+r(x)$?

    The remainder polynomial $r(x)$. What's left after division when $\deg(r)<\deg(b)$.

43. What is $\frac{x^3-1}{x-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x^2+x+1$. Factor $x^3-1=(x-1)(x^2+x+1)$ then cancel $(x-1)$.

44. What is the key stopping condition in polynomial long division?

    Stop when $\deg(\text{remainder})<\deg(b(x))$. Division ends when remainder degree drops below divisor degree.

45. What is $\frac{x^2+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x-1+\frac{2}{x+1}$. Long division gives quotient $x-1$ with remainder $2$.

46. What must you do first before long dividing by $x^2+3$ if terms are missing in $a(x)$?

    Insert missing powers with coefficient $0$. Ensures all terms are present for proper column alignment.

47. What is $\frac{2x^2+3x+1}{x+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $2x+1$. Factor as $ (x+1)(2x+1) $ then cancel $ (x+1) $.

48. What is $\frac{x^3-x}{x^2-1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x$. Factor as $x(x^2-1)=x(x+1)(x-1)$ then simplify.

49. What is $\frac{5x^3-10x}{5x}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x^2-2$. Split each term: $\frac{5x^3}{5x}-\frac{10x}{5x}=x^2-2$.

50. What is the identity that connects division form and remainder form for polynomials?

    $a(x)=b(x)q(x)+r(x)$. The fundamental division algorithm for polynomials.

51. What is $\frac{x^3+1}{x^2+1}$ written as $q(x)+\frac{r(x)}{b(x)}$?

    $x+\frac{-x+1}{x^2+1}$. Long division gives quotient $x$ with remainder $-x+1$.

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