Rewriting Rational Expressions - Algebra 2

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

   $x^2+x+1+\frac{0}{x-1}$. Use identity $x^3-1=(x-1)(x^2+x+1)$ for exact division.

2. What is $\frac{6x^2+7x-3}{3x+1}$ rewritten as $q(x)+\frac{r(x)}{3x+1}$?

   $2x+\frac{5x-3}{3x+1}$. Divide quadratic by linear expression $3x+1$.

3. What identity rewrites $a(x)$ after division by $b(x)$?

   $a(x)=b(x)q(x)+r(x)$. Division algorithm identity relating dividend, divisor, quotient, and remainder.

4. What must you do if $a(x)$ is missing an $x^2$ term before long division?

   Insert $0x^2$ as a placeholder. Missing terms need zero coefficients for proper alignment.

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

   $x^2-2x+1+\frac{0}{x-1}$. Perfect cube $ $(x-1)^3$ $ divided by $ (x-1) $.

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

   $x+1+\frac{-x-1}{x^2+1}$. Factor $x^2$ and divide by $x^2+1$.

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

   $x+1+\frac{0}{x-1}$. Factor $x^2-1=(x-1)(x+1)$ for exact division.

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

   $x-1+\frac{0}{x+1}$. Factor $x^2-1=(x+1)(x-1)$ for exact division.

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

   $x+\frac{-x}{x^2+1}$. Divide cubic by quadratic to get linear quotient.

10. Identify the correct rewrite: $\frac{a(x)}{b(x)}$ equals what expression using $q(x)$ and $r(x)$?

    $q(x)+\frac{r(x)}{b(x)}$. Standard polynomial division form.

11. What condition must $r(x)$ satisfy 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.

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

    $3x+1+\frac{5}{x-1}$. Polynomial division with quotient degree 1.

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

    $x^2+x+\frac{1}{x+1}$. Long division of cubic by linear expression.

14. What is the remainder theorem statement for dividing by $x-c$?

    Remainder is $a(c)$. Evaluating the polynomial at $c$ gives the remainder.

15. What form results from dividing polynomials: $\frac{a(x)}{b(x)}= ?$

    $q(x)+\frac{r(x)}{b(x)}$. Quotient plus remainder over divisor form.

16. What is a quick method (instead of long division) for dividing by $x-c$?

    Synthetic division. Efficient method for linear divisors.

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

    $x+1+\frac{4}{x+1}$. Long division yields quotient $x+1$ and remainder $4$.

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

    $x^2+2x+\frac{3}{x^2}$. Divide each term of the numerator by $x^2$.

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

    $x^2+\frac{2}{x-1}$. Divide cubic by linear to get quadratic quotient.

20. What must be true about $a(x)$ and $b(x)$ to start long division of $\frac{a(x)}{b(x)}$?

    Write both in descending powers, include $0$ terms. Standard form needed for the division algorithm.

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

    $x^2+2x+4+\frac{0}{x-2}$. Use identity $x^3-8=(x-2)(x^2+2x+4)$ for exact division.

22. What is $\frac{x^3+1}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

    $x^2-x+1+\frac{0}{x+1}$. Use identity $x^3+1=(x+1)(x^2-x+1)$ for exact division.

23. What form results from dividing polynomials: $\frac{a(x)}{b(x)}=\ ?$

    $q(x)+\frac{r(x)}{b(x)}$. Quotient plus remainder over divisor form.

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

    Polynomial long division. Standard algorithm for dividing polynomials.

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

    $x+2+\frac{-2x+1}{x^2+1}$. Divide cubic by quadratic using long division.

26. What is $\frac{x^3+2x^2+4x+8}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?

    $x^2+4+\frac{0}{x+2}$. Factor by grouping: $(x+2)(x^2+4)$ for exact division.

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

    $x+2+\frac{0}{x-2}$. Factor $x^2-4=(x-2)(x+2)$ for exact division.

28. What is $\frac{x^3+4}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

    $x^2-x+1+\frac{3}{x+1}$. Divide cubic by linear using long division.

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

    $2x+1+\frac{2}{2x-1}$. Divide by the linear expression $2x-1$.

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

    $2x+7+\frac{7}{x-2}$. Polynomial division with quadratic quotient.

31. What is $r(x)$ when $b(x)$ divides $a(x)$ evenly in $\frac{a(x)}{b(x)}$?

    $r(x)=0$. No remainder when division is exact.

32. What is the quotient called when dividing by $x-c$ using synthetic division?

    The depressed polynomial $q(x)$. The quotient after factoring out $(x-c)$.

33. What is $\frac{x^2+4x+7}{x-1}$ rewritten as $q(x)+\frac{r(x)}{x-1}$?

    $x+5+\frac{12}{x-1}$. Long division with quotient $x+5$ and remainder $12$.

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

    $x+\frac{-3x}{x^2-1}$. Factor out $x$ first: $x^3-4x=x(x^2-4)$.

35. What is the first step to find the leading term of $q(x)$ in long division?

    Divide leading term of $a(x)$ by leading term of $b(x)$. Divide highest degree terms first.

36. What is the degree of the quotient when dividing degree $n$ by degree $m$ with $n\ge m$?

    $\deg(q(x))=n-m$. Quotient degree equals dividend minus divisor degree.

37. What is $\frac{x^2-4x+7}{x+1}$ rewritten as $q(x)+\frac{r(x)}{x+1}$?

    $x-5+\frac{12}{x+1}$. Divide quadratic by $x+1$ using polynomial division.

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

    $x+\frac{1}{x}$. Divide to get quotient $x$ and remainder $1$.

39. What is $q(x)$ when $\deg(a(x))<\deg(b(x))$ for $\frac{a(x)}{b(x)}$?

    $q(x)=0$. No quotient when dividend degree is smaller.

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

    $x-1+\frac{4}{x+1}$. Divide $x^2+3$ by $x+1$ using polynomial long division.

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

    $x-2+\frac{0}{x+2}$. Factor $x^2-4=(x+2)(x-2)$ for exact division.

42. What is $\frac{x^4+x}{x^2}$ rewritten as $q(x)+\frac{r(x)}{x^2}$?

    $x^2+\frac{1}{x}$. Simplify by dividing each term by $x^2$.

43. What is $\frac{x^3-2x^2+4x-8}{x-2}$ rewritten as $q(x)+\frac{r(x)}{x-2}$?

    $x^2+4+\frac{0}{x-2}$. Factor by grouping: $(x-2)(x^2+4)$ for exact division.

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

    $x+3+\frac{-9}{x+3}$. Rewrite as $\frac{x(x+6)}{x+3}$ and divide.

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

    $x-1+\frac{-x+1}{x^2+x}$. Divide cubic by quadratic $x^2+x=x(x+1)$.

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

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

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

    $x+2+\frac{0}{x+1}$. Factor $x^2+3x+2=(x+1)(x+2)$ for exact division.

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

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

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

    $2x+3+\frac{x+8}{x^2-1}$. Long division of cubic by quadratic expression.

50. What is $\frac{x^3+8}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?

    $x^2-2x+4+\frac{0}{x+2}$. Use identity $x^3+8=(x+2)(x^2-2x+4)$ for exact division.

51. What is the remainder when $a(x)$ is divided by $x+3$?

    $a(-3)$. Apply remainder theorem with $x+3=(x-(-3))$.

52. What is the remainder when $a(x)$ is divided by $x$?

    $a(0)$. Substitute $x=0$ using the remainder theorem.

53. What is $\frac{5x^2+10x+3}{5x+5}$ rewritten as $q(x)+\frac{r(x)}{5x+5}$?

    $x+1+\frac{-2}{5x+5}$. Factor out 5 and divide: $\frac{5(x^2+2x)+3}{5(x+1)}$.

54. What is $\frac{x^3+4x^2+5x+2}{x+2}$ rewritten as $q(x)+\frac{r(x)}{x+2}$?

    $x^2+2x+1+\frac{0}{x+2}$. Exact division since $(x+2)$ is a factor.

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

    $x^2-1+\frac{0}{x^2+1}$. Use identity $x^4-1=(x^2+1)(x^2-1)$ for exact division.

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

    $x+3+\frac{9}{x-3}$. Add and subtract 9: $x^2=(x-3)(x+3)+9$.

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