Table of Contents
What are Partial Fractions?
A partial fraction is a mathematical expression that is used to represent the quotient of two polynomials. The denominator of the fraction is the polynomial that is divided by the other polynomial. The numerator of the fraction is the polynomial that is left over after the division has been completed.
Partial Fractions from Rational Functions
A rational function is a function that can be expressed as the quotient of two polynomials. The denominator of a rational function is always a polynomial. The numerator can be a polynomial, or it can be a rational function.
To find the partial fractions of a rational function, we first need to simplify the function. We can do this by factoring the denominator and cancelling out any common factors.
Once the function is simplified, we can divide the numerator and denominator by the greatest common factor (GCF). This will give us the simplest form of the rational function.
The partial fractions of a rational function are the fractions that make up the simplified function. We can write these fractions in terms of the original function’s variables.
We can also use the partial fractions to find the inverse of a rational function. To do this, we first need to solve for the roots of the denominator. We can then use these roots to find the inverse of the rational function.
Proper Fraction of Improper Function
The improper fraction of an improper function is the improper fraction of the function’s argument.
Here’s How to Solve Partial Fractions!
To solve partial fractions, we need to decompose the given fraction into a series of simpler fractions.
We can do this by using the algebraic method, or the graphical method.
Algebraic Method
1. Decompose the given fraction into a series of simpler fractions.
2. Write out the algebraic equation for each of the fractions.
3. Solve each of the equations.
4. Combine the solutions to get the final answer.
Graphical Method
1. Draw a graph of the given fraction.
2. Find the points of intersection of the graph and the x-axis.
3. Find the points of intersection of the graph and the y-axis.
4. Write out the algebraic equation for each of the fractions.
5. Solve each of the equations.
6. Combine the solutions to get the final answer.
Now, What are Proper Rational Expressions?
A proper rational expression is a rational expression in which the numerator and denominator are polynomials.
What is the Degree?
The degree is a Master of Arts in Teaching.
Partial Fractions Decomposition
\[\frac{x}{x^{2}-4}=\frac{A}{x-2}+\frac{B}{x-4}+\frac{C}{x^{2}-4}\]
We can rewrite this equation as:
\[x=A(x-2)+B(x-4)+C(x^{2}-4)\]
We can then use the distributive property to simplify this equation:
\[x=Ax-2+Bx-4+Cx^{2}-4\]
We can then combine like terms:
\[x=Ax-2+Bx-4+Cx^{2}-4=Ax-2+Bx-4-Cx^{2}+4\]
We can then move the -Cx^{2} term to the right side of the equation:
\[x-Cx^{2}=4\]
We can then divide both sides of the equation by -Cx^{2}:
\[x=\frac{4}{-Cx^{2}}\]
Partial Fractions Examples and Solutions (Integration)
1) $\frac{x^2}{4} + \frac{x}{4}$
We can rewrite the given equation as:
$\frac{x^2}{4} = \frac{x}{4} + \frac{1}{4}$
Now, we can use the standard formula for integrating fractions:
$\int \frac{x}{4} dx = \frac{1}{4}x^2 + C$
2) $\frac{x^3}{27} + \frac{x}{9}$
We can rewrite the given equation as:
$\frac{x^3}{27} = \frac{x}{9} + \frac{3}{9}$
Now, we can use the standard formula for integrating fractions:
$\int \frac{x}{9} dx = \frac{3}{9}x^3 + C$
Tips To Split A Fraction into its Partial Fractions
There are a few steps in order to split a fraction into its partial fractions.
1. Decide which denominator to begin with. In some cases, all of the denominators may be the same.
2. Write the original fraction as a fraction with the chosen denominator.
3. Factor the original fraction’s denominator.
4. Write each factor of the denominator as a fraction with the chosen denominator.
5. Add the fractions with the chosen denominators.
6. Write the final answer as a fraction with the original denominator.
