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- Rational Expressions

Still Confused?

Try reviewing these fundamentals first

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Try reviewing these fundamentals first

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Get Started Now- Intro Lesson9:49
- Lesson: 1a16:29
- Lesson: 1b7:49
- Lesson: 2a14:50
- Lesson: 2b6:36
- Lesson: 38:37
- Lesson: 418:13
- Lesson: 5a12:10
- Lesson: 5b11:31

Basic Concepts:Integration of rational functions by partial fractions,

$\bullet$ 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.

- IntroductionIntroduction to Partial Fraction Decompositiona)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)$\frac{x + 7}{(x + 3)(x - 1)}$b)$\frac{4x + 3}{x^{2} + x}$ - 2.
**Case 2: Denominator is a product of linear factors with repeats**Find the partial fractions of :

a)$\frac{3x^{2} - 5}{(x - 2)^{3}}$b)$\frac{2x - 1}{x^{2} + 10x + 25}$ - 3.
**Case 3: Denominator contains irreducible quadratic factors with no repeats**Find the partial fractions of :

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

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

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

- 5.
**First perform long division, then partial fraction decomposition**Find the partial fractions of:

a)$\frac{x^{3} - 3x^{2} + 4x}{x^{2} - 3x 2}$b)$\frac{2x^{2} + 14x + 24}{x^{2} + 6x - 16}$

5.

Rational Expressions

5.1

Simplifying rational expressions and restrictions

5.2

Adding and subtracting rational expressions

5.3

Multiplying rational expressions

5.4

Dividing rational expressions

5.5

Solving rational equations

5.6

Applications of rational equations

5.7

Simplifying complex fractions

5.8

Partial fraction decomposition