Using Linear Partial Fractions - AP Calculus BC

1. What is the result of using partial fractions on $\frac{1}{(x+1)^2}$?

   $\frac{A}{x+1} + \frac{B}{(x+1)^2}$. Repeated factor already factored, so direct decomposition.

2. Determine the partial fractions for $\frac{8x}{x^2-2x}$.

   $\frac{A}{x} + \frac{B}{x-2}$. Factor $x^2-2x = x(x-2)$.

3. What is the first step in decomposing a rational function into partial fractions?

   Factor the denominator. Essential for setting up partial fraction terms.

4. Identify the partial fractions for $\frac{7x+9}{x^2+x}$.

   $\frac{A}{x} + \frac{B}{x+1}$. Factor $x^2+x = x(x+1)$.

5. What is the purpose of equating coefficients in partial fraction decomposition?

   To solve for unknown constants. Matching coefficients determines constants.

6. What is the advantage of using partial fractions in integration?

   Simplifies the integral into manageable parts. Each simple fraction integrates easily.

7. What does each term in a partial fraction decomposition represent?

   A simpler fraction with a linear denominator. Each term has constant numerator and linear denominator.

8. Find $A$ in the decomposition $\frac{A}{x} + \frac{B}{x-1} = \frac{2x+3}{x(x-1)}$.

   $A = 2$. Set $x=0$ to find $A$.

9. Calculate $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{4}{x(x+1)}$.

   $B = -4$. Set $x=-1$ to isolate $B$.

10. What must be true about the degree of the numerator relative to the denominator?

    Degree of numerator < degree of denominator. Proper fractions are required for decomposition.

11. Identify the partial fraction form for $\frac{1}{(x-1)(x-2)(x-3)}$.

    $\frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x-3}$. Three distinct linear factors need three terms.

12. Find the partial fraction decomposition of $\frac{4x+2}{x^2-4}$.

    $\frac{A}{x-2} + \frac{B}{x+2}$. Factor $x^2-4 = (x-2)(x+2)$.

13. What substitution is used to solve for $A$ in partial fractions?

    Set $x$ to make other term zero. Strategic substitution eliminates other terms.

14. What is the partial fraction form for a simple linear factor like $x-a$?

    $\frac{A}{x-a}$. Each linear factor gets one partial fraction term.

15. What is a key benefit of partial fraction decomposition for solving integrals?

    Breaks down complex rational functions. Makes integration much simpler.

16. Identify the partial fractions for $\frac{x^2+3x+2}{x^3-x}$.

    $\frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1}$. Factor $x^3-x = x(x-1)(x+1)$.

17. Find the partial fractions for $\frac{x+5}{x^2+5x+6}$.

    $\frac{A}{x+2} + \frac{B}{x+3}$. Factor $x^2+5x+6 = (x+2)(x+3)$.

18. What is the general form of a linear partial fraction decomposition?

    $\frac{A}{x-a} + \frac{B}{x-b}$. For distinct linear factors, each gets one term.

19. What is the form of partial fractions for a factor like $(x-a)^n$?

    $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + ... + \frac{A_n}{(x-a)^n}$. Each power of repeated factor needs its own term.

20. Identify the partial decomposition for $\frac{3x+1}{(x-3)^2}$.

    $\frac{A}{x-3} + \frac{B}{(x-3)^2}$. Repeated factor requires two terms.

21. State the partial fraction form for $\frac{6x+9}{x^3(x-1)}$.

    $\frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{D}{x-1}$. $x^3$ factor creates three terms plus linear factor.

22. What is the purpose of partial fraction decomposition in calculus?

    To simplify integration or differentiation. Converts complex fractions into simpler integrable forms.

23. Determine $A$ in the decomposition $\frac{A}{x} + \frac{B}{x+1} = \frac{3}{x(x+1)}$.

    $A = 3$. Set $x=0$ to isolate $A$.

24. Identify the partial fractions for $\frac{5x}{x^2+3x}$.

    $\frac{A}{x} + \frac{B}{x+3}$. Factor out $x$ from denominator first.

25. What must be done if the rational function is improper for partial fractions?

    Perform polynomial long division first. Convert improper to proper fraction first.

26. Find the partial fractions for $\frac{x+6}{x^2-5x+6}$.

    $\frac{A}{x-2} + \frac{B}{x-3}$. Factor $x^2-5x+6 = (x-2)(x-3)$.

27. Determine the partial fractions for $\frac{7x+4}{x^2-1}$.

    $\frac{A}{x-1} + \frac{B}{x+1}$. Factor $x^2-1 = (x-1)(x+1)$.

28. Explain why partial fractions cannot be used if the fraction is improper.

    Numerator degree must be less than denominator. Improper fractions need polynomial division first.

29. Identify the partial fraction decomposition for $\frac{1}{x(x+1)}$.

    $\frac{A}{x} + \frac{B}{x+1}$. Factor denominator $x(x+1)$, then assign constants.

30. Identify the partial fractions for $\frac{2x+1}{x^2+x-6}$.

    $\frac{A}{x-2} + \frac{B}{x+3}$. Factor $x^2+x-6 = (x-2)(x+3)$.

31. Find the partial fraction decomposition of $\frac{2x+3}{(x-1)(x+2)}$.

    $\frac{A}{x-1} + \frac{B}{x+2}$. Factor $(x-1)(x+2)$, assign constants to each.

32. What is the form of partial fractions for a repeated linear factor like $(x-a)^2$?

    $\frac{A}{x-a} + \frac{B}{(x-a)^2}$. Repeated factors need terms for each power.

33. What is the denominator structure required for partial fraction decomposition?

    Product of linear or irreducible quadratic factors. Standard form required for partial fraction method.

34. What is the benefit of using partial fractions in solving integrals?

    Facilitates integration by breaking into simpler terms. Each partial fraction integrates using basic rules.

35. Determine the partial fractions for $\frac{3x^2}{x^3-x}$.

    $\frac{A}{x} + \frac{B}{x-1} + \frac{C}{x+1}$. Factor $x^3-x = x(x-1)(x+1)$.

36. Identify the partial fractions for $\frac{5}{x^2-4x+4}$.

    $\frac{A}{x-2} + \frac{B}{(x-2)^2}$. Factor $x^2-4x+4 = (x-2)^2$.

37. Find $B$ in the decomposition $\frac{A}{x} + \frac{B}{x+2} = \frac{2x+5}{x(x+2)}$.

    $B = 5$. Set $x=-2$ to isolate $B$.

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