What are Partial Fractions?
Partial Fractions are the simpler fractions you get when you break up a more complex rational function/expression.
What is the Partial Fraction Formula?
Here’s how partial fraction formulas works:
- Start with a fraction that has polynomials on top and bottom.
- Factor the bottom (denominator) completely.
- For each factor in the denominator, you create a new, simpler fraction.
- If a factor repeats, you include it multiple times with increasing powers.
- Add all these simple fractions together.
The formula looks like this: Original Fraction = A / (x + a) + B / (x + b) + (Cx + D) / (x² + ex + f) + …Where A, B, C, D, etc. are numbers you need to figure out.
What is Partial Fraction Expansion?
Partial fraction expansion is a method of decomposing a complicated fraction into a sum of simpler fractions. It’s like breaking a big, difficult problem into smaller, more manageable pieces. This process is also known as partial fraction decomposition.
For example:
You can say that the right hand side are the partial fractions of the left hand side.
Before you can break up a rational expression into its partial fractions, you need to learn…
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Terminology
Proper and Improper Algebraic Fractions
Given an algebraic fraction f(x)/g(x),
Proper Fraction
The algebraic fraction is called a proper fraction when the degree of the numerator f(x) is less than the degree of the denominator g(x)
Improper Fraction
The algebraic fraction is called an improper fraction when the degree of the numerator f(x) is equal to or more than the degree of the denominator g(x).
Denominator Types
Distinct Linear Factor
Distinct linear factors are linear terms (ax + b) that appear only once in the denominator of a fraction. For example, in the fraction (x + 2) / ((x – 1)(x + 3)), (x – 1) and (x + 3) are distinct linear factors.
Repeated Linear Factor
Repeated linear factors refer to when a linear term in the denominator of a rational expression occurs more than once. For example, in the fraction (x + 1) / (x – 2)2, (x – 2) is a repeated linear factor because it appears twice.
Irreducible Quadratic Factor
An irreducible quadratic factor is a quadratic expression in the denominator of a rational function that cannot be factored further over the real numbers.
Expressing an Algebraic Fraction as a Sum of its Partial Fractions
Step 1: Ensure that the algebraic fraction is proper. If the algebraic fraction is improper, carry out long division and then express it as a sum of a polynomial and a proper fraction.
Step 2: For the proper algebraic fraction, ensure that its denominator is completely factorised.
Step 3: Apply the rules in the following table to rewrite the proper fraction as a sum of its partial fractions.
Case 1: The denominator g(x) has Distinct linear factors in the form: ax or (ax + b)
For each factor in the denominator, assign corresponding partial fraction(s) as follow:
Example:
Case 2: The denominator g(x) has Repeated linear factors in the form: (ax)2 or (ax + b)2
For each factor in the denominator, assign corresponding partial fraction(s) as follow:
Examples:
Case 3: The denominator g(x) has Irreducible quadratic factors in the form: x2 + c2
*An irreducible factor is a factor that cannot be factorised any further.
For each factor in the denominator, assign corresponding partial fraction(s) as follow:
Examples:
Step 4: Drop the all denominators by both multiplying both sides of the equation by the LCM of the denominators. An algebraic identity is then obtained.
Step 5: Find the values of all the unknown constants in the identity by using any of the methods below.
Method 1: Substitute certain values of x into the identity.
Method 2: Comparing coefficients on both sides of the identity.
Example of Partial Fractions without Long Division
Express the following in partial fraction
Solution: We first check whether the fraction is a proper fraction. Since the degree of the numerator (2) is less than the degree of the denominator (3), we can conclude that this is a proper fraction and we can skip the long division process.
x2−x+10 =A(x2+3) + (Bx+C)(x+1)
Let x = −1
12 = 4A
A = 3
Let x = 0 (also same as comparing constants).
10 = 3A + C
C = 10 – 3(3) = 1
Comparing coefficients of x2:
1 = A + B
B = 1 – 3 = −2
Hence
OR
Example of Partial Fractions with Long Division
Express the following in partial fractions.
Solution: We first check whether the fraction is a proper fraction. Since the degree of the numerator (3) is more than the degree of the denominator (2), we can conclude that this is an improper fraction and we need to do long division first.
We first expand the denominator. (x + 3)(x – 1) = x2 + 2x − 3
Long Division:
We can now express the improper fraction in this format:
For example:
16 = 3 x 5 + 1,
16/3 = 5 + 1/3 .
Therefore,
We now have a proper fraction on the right-hand side and can proceed to break it into its partial fractions.
Let
x – 5 = A(x – 1) + B(x + 3)
Let x = −3
−8 = −4A
A = 2
Let x = 1
−4 = 4B
B = −1
Hence,
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