# Partial fraction decomposition

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##### Intros
###### Lessons
1. Introduction to Partial Fraction Decomposition
What is partial fraction decomposition?
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##### Examples
###### Lessons
1. Case 1: Denominator is a product of linear factors with no repeats

Find the partial fractions of:

1. $\frac{x + 7}{(x + 3)(x - 1)}$
2. $\frac{4x + 3}{x^{2} + x}$
2. Case 2: Denominator is a product of linear factors with repeats

Find the partial fractions of :

1. $\frac{3x^{2} - 5}{(x - 2)^{3}}$
2. $\frac{2x - 1}{x^{2} + 10x + 25}$
3. Case 3: Denominator contains irreducible quadratic factors with no repeats

Find the partial fractions of :

$\frac{2x^{2} + 5x + 8}{x^{3} - 8x}$

1. Case 4: Denominator contains irreducible quadratic factors with repeats

Find the partial fractions of:

$\frac{3x^{4} + x^{3} + 1}{x(x^{2} + 1)^{2}}$

1. First perform long division, then partial fraction decomposition

Find the partial fractions of:

1. $\frac{x^{3} - 3x^{2} + 4x}{x^{2} - 3x 2}$
2. $\frac{2x^{2} + 14x + 24}{x^{2} + 6x - 16}$
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###### Topic Notes

$\bullet$ Partial fraction decomposition expresses a rational function $\frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are polynomials in $x$, as a sum of simpler fractions.

$\bullet$Partial fraction decomposition only applies to proper fractions in which the degree of the numerator is less than that of the denominator.