FINLANDIA SCHOOLA partial fraction is one of the simpler fractions obtained when a rational algebraic expression is decomposed into two or more simpler fractions.
For example:
can be written as:
The process of changing one complicated rational expression into simpler fractions is called partial fraction decomposition.
8. Proper Rational Functions
A rational function has the form:
where � and � are polynomials and �.
It is proper when:
Example
is proper because:
Numerator degree = 1
Denominator degree = 2
Therefore:
Improper example
is improper because the numerator has degree 2 while the denominator has degree 1.
Important point
For an improper rational function, perform polynomial division first before decomposing into partial fractions.
9. Distinct Linear Factors
Consider:
The denominator contains two distinct linear factors:
Therefore, write:
Step 1: Multiply through by the denominator
Step 2: Choose convenient values
Let:
Then:
Now let:
Therefore:
Teacher Tip
Tell students:
"When the factors are distinct linear factors, give each factor its own constant numerator."
10. Repeated Linear Factors
Consider:
The denominator contains a repeated factor:
Therefore, the correct form is:
Example
Decompose:
Write:
Multiply by �:
Put �:
Using comparison of coefficients:
Therefore:
So:
11. Repeated Factors – General Form
If the denominator contains:
the partial fractions must include every power:
For:
we need:
Key rule
Do not skip any power of a repeated factor.
12. Other Denominator Forms
A. Irreducible Quadratic Factor
If the denominator contains a quadratic factor that cannot be factorised further over the real numbers, such as:
the numerator above it must be linear.
For example:
Use:
Notice that the numerator over the quadratic factor is:
not just a constant.
13. Mixed Denominator Example
Consider:
The denominator contains:
One distinct linear factor: �
One repeated linear factor: �
Therefore, the correct setup is:
Important examination rule
Students should factorise the denominator completely before writing the partial fractions.