Types of Partial Fractions

Question 1

1. Convert expression with two linear factors in denominator and constant in numerator. e.g.
   f(x) = A/[(ax+b)(cx+d)]

Answer 1

Split into two fractions with the denominators being the original factors …
f(x) = A/(ax+b) + B/(cx+d)
Then use substitution to solve for A,B – use roots where possible, and then simple values to get more equations in A, B
Use coefficient check, as well as a random substitution check…

Question 2

1. Expression with 3 or more linear terms in denominator and constant numerator
   f(x) = k/[(ax+b)(cx+d)(ex+f)…]

Answer 2

f(x) = A/(ax+b) + B/(cx+d) + C/(ex+f)….

• Substitution to form equations in, and solve for A,B,C, using roots where possible
• Coefficient check, random sub check

Question 3

2. REPEATED linear terms in denominator and with numerator of degree less than denominator…
   e. g. f(x) = (Px^2 + Qx + R)/[(ax+b)^2 (cx+d)]

Answer 3

Partial fractions need denominators which account for ‘all possible’ denominators. Repeated factor must be present in two PFs as a denominator and as a ‘repeated’ denominator
f(x) = A/(ax+b) + B/(ax+b)^2 + C/(cx+d)
- Substitute, solve, using roots where possible
- Coefficient and random sub checks.

Question 4

3 IMPROPER FRACTIONS. Types (1 Linear factors in denominator) and (2 repeated linear factors in denominator) ARE ONLY SUFFICIENT WHEN THE DEGREE OF THE NUMERATOR is LESS THAN the degree of the denominator. What extra thing do you have to do when the degree of the numerator is EQUAL TO or GREATER THAN the degree of the denominator?
e.g.
f(x) = (x^2 + 2x + 4)/(x^2 + 2x + 2) ?

Answer 4

• First do POLYNOMIAL LONG DIVISION to get the expression in the form of [expression + proper fraction).
  – Then apply techniques 1 and 2 to convert the proper fraction into partial fractions, using appropriate checking.
  nb proper fraction has degree of numerator LESS THAN degree of denominator

Question 5

What are the three “types of partial fraction” techniques?

Answer 5

1. Linear factors in denominator, PROPER FRACTION (numerator with degree less than degree of denominator)
2. Repeated linear factors in denominator, PROPER FRACTION (numerator with degree less than degree of denominator).
3. IMPROPER FRACTION (either of above) where degree of numerator is equal to or exceeds the degree of the denominator.