Partial fractions questions with answers

Questions

1. Question 1Two linear factors

   Express $\dfrac{5x - 11}{(x - 3)(x - 1)}$ in partial fractions.

   Show the worked solution

   1. Step 1: One fraction for each linear factor$\dfrac{5x - 11}{(x - 3)(x - 1)} = \dfrac{A}{x - 3} + \dfrac{B}{x - 1}$
   2. Step 2: Multiply both sides by the denominator$5x - 11 = A(x - 1) + B(x - 3)$
   3. Step 3: Find A: substitute the value of x that makes the other terms vanish$5(3) - 11 = A(3 - 1)$$4 = 2A$$A = 2$
   4. Step 4: Find B: substitute the value of x that makes the other terms vanish$5(1) - 11 = B(1 - 3)$$-6 = -2B$$B = 3$So it is $\dfrac{2}{x - 3} + \dfrac{3}{x - 1}$.
2. Question 2A repeated factor

   Express $\dfrac{3x^2 + 9x - 3}{(x - 1)(x + 2)^2}$ in partial fractions.

   Show the worked solution

   1. Step 1: Write it out with the factors of the denominator$\dfrac{3x^2 + 9x - 3}{(x - 1)(x + 2)^2} = \dfrac{A}{x - 1} + \dfrac{B}{x + 2} + \dfrac{C}{(x + 2)^2}$
   2. Step 2: Multiply both sides by the denominator$3x^2 + 9x - 3 = A(x + 2)^2 + B(x - 1)(x + 2) + C(x - 1)$
   3. Step 3: Find A: substitute the value of x that makes the other terms vanish$3(1)^2 + 9(1) - 3 = A(1 + 2)^2$$9 = 9A$$A = 1$
   4. Step 4: Find C: substitute the value of x that makes the other terms vanish$3(-2)^2 + 9(-2) - 3 = C(-2 - 1)$$-9 = -3C$$C = 3$
   5. Step 5: Compare the coefficients of x²$A + B = 3$$B = 2$So it is $\dfrac{1}{x - 1} + \dfrac{2}{x + 2} + \dfrac{3}{(x + 2)^2}$.
3. Question 3A quadratic factor

   Express $\dfrac{3x^2 + x + 16}{(x + 2)(x^2 + 9)}$ in partial fractions.

   Show the worked solution

   1. Step 1: Write it out with the factors of the denominator$\dfrac{3x^2 + x + 16}{(x + 2)(x^2 + 9)} = \dfrac{A}{x + 2} + \dfrac{Bx + C}{x^2 + 9}$
   2. Step 2: Multiply both sides by the denominator$3x^2 + x + 16 = A(x^2 + 9) + (Bx + C)(x + 2)$
   3. Step 3: Find A: substitute the value of x that makes the other terms vanish$3(-2)^2 + (-2) + 16 = A((-2)^2 + 9)$$26 = 13A$$A = 2$
   4. Step 4: Compare the coefficients of x²$A + B = 3$$B = 1$
   5. Step 5: Compare the coefficients of x$C + 2B = 1$$C = -1$So it is $\dfrac{2}{x + 2} + \dfrac{x - 1}{x^2 + 9}$.
4. Question 4Improper fractions: divide first

   Express $\dfrac{x^{3} - x^{2} - 7x + 1}{(x - 1)(x - 3)}$ in partial fractions.

   Show the worked solution

   1. Step 1: The top's degree is not less than the bottom's: expand the bottom and divide first$(x - 1)(x - 3) = x^{2} - 4x + 3$
   2. Step 2: Divide: quotient + remainder over the bottom$\dfrac{x^{3} - x^{2} - 7x + 1}{(x - 1)(x - 3)} = x + 3 + \dfrac{2x - 8}{(x - 1)(x - 3)}$
   3. Step 3: Split the proper fraction: cover up each factor (or compare)$\dfrac{2x - 8}{(x - 1)(x - 3)} = \dfrac{A}{x - 1} + \dfrac{B}{x - 3}$$A = 3$$B = -1$
   4. Step 4: Put it together$x + 3 + \dfrac{3}{x - 1} - \dfrac{1}{x - 3}$