Partial fraction decomposition - Rational Equations and Expressions

Partial fraction decomposition

Basic concepts:
Factor by taking out the greatest common factor
Factoring trinomials

Related concepts:
Integration of rational functions by partial fractions

Lessons

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.

- a)
  What is partial fraction decomposition?
- b)
  When can we perform partial fraction decomposition?
1.
Case 1: Denominator is a product of linear factors with no repeats
Find the partial fractions of:
2.
Case 2: Denominator is a product of linear factors with repeats
Find the partial fractions of :
5.
First perform long division, then partial fraction decomposition
Find the partial fractions of:

Partial fraction decomposition

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Factoring trinomials

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Lessons

Notes:

Intro Lesson

Introduction to Partial Fraction Decomposition

a)

What is partial fraction decomposition?

b)

When can we perform partial fraction decomposition?

1.

Case 1: Denominator is a product of linear factors with no repeats Find the partial fractions of:

a)

x+7(x+3)(x−1)\frac{x + 7}{(x + 3)(x - 1)}(x+3)(x−1)x+7​

b)

4x+3x2+x\frac{4x + 3}{x^{2} + x}x2+x4x+3​

2.

Case 2: Denominator is a product of linear factors with repeats Find the partial fractions of :

a)

3x2−5(x−2)3\frac{3x^{2} - 5}{(x - 2)^{3}}(x−2)33x2−5​

b)

2x−1x2+10x+25\frac{2x - 1}{x^{2} + 10x + 25}x2+10x+252x−1​

3.

Case 3: Denominator contains irreducible quadratic factors with no repeats Find the partial fractions of : 2x2+5x+8x3−8x\frac{2x^{2} + 5x + 8}{x^{3} - 8x}x3−8x2x2+5x+8​

4.

Case 4: Denominator contains irreducible quadratic factors with repeats Find the partial fractions of: 3x4+x3+1x(x2+1)2\frac{3x^{4} + x^{3} + 1}{x(x^{2} + 1)^{2}}x(x2+1)23x4+x3+1​

5.

First perform long division, then partial fraction decomposition Find the partial fractions of:

a)

x3−3x2+4xx2−3x2\frac{x^{3} - 3x^{2} + 4x}{x^{2} - 3x 2}x2−3x2x3−3x2+4x​

b)

2x2+14x+24x2+6x−16\frac{2x^{2} + 14x + 24}{x^{2} + 6x - 16}x2+6x−162x2+14x+24​

Partial fraction decomposition

Don't just watch, practice makes perfect.

We have over 1850 practice questions in Algebra for you to master.

Algebra
Factor by taking out the greatest common factor

Algebra
Factoring trinomials

or

Case 1: Denominator is a product of linear factors with no repeats
Find the partial fractions of:

$\frac{x + 7}{(x + 3)(x - 1)}$

$\frac{4x + 3}{x^{2} + x}$

Case 2: Denominator is a product of linear factors with repeats
Find the partial fractions of :

$\frac{3x^{2} - 5}{(x - 2)^{3}}$

$\frac{2x - 1}{x^{2} + 10x + 25}$

Case 3: Denominator contains irreducible quadratic factors with no repeats
Find the partial fractions of :

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

Case 4: Denominator contains irreducible quadratic factors with repeats
Find the partial fractions of:

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

First perform long division, then partial fraction decomposition
Find the partial fractions of:

$\frac{x^{3} - 3x^{2} + 4x}{x^{2} - 3x 2}$

$\frac{2x^{2} + 14x + 24}{x^{2} + 6x - 16}$