Lesson
Practice Problems
Decompose the following fractions.
\(\textbf{1)}\) \(\displaystyle\frac{8x+10}{x^2+2x} \)
\(\textbf{2)}\) \(\displaystyle\frac{4x+8}{x^2+6x+5} \)
\(\textbf{3)}\) \(\displaystyle\frac{1}{x^2-5x+6} \)
\(\textbf{4)}\) \(\displaystyle\frac{9x+9}{2x^2+11x+5} \)
\(\textbf{5)}\) \(\displaystyle\frac{49x^2}{(x+3)(x-4)^2} \)
\(\textbf{6)}\) \(\displaystyle\frac{2x+5}{(x+5)^2} \)
\(\textbf{7)}\) \(\displaystyle\frac{6x^2-3}{x(x^2+3x+3)} \)
\(\textbf{8)}\) \(\displaystyle\frac{x^3+4x+5}{(x^2+3)^2} \)
See Related Pages\(\)
\(\bullet\text{ Partial Fraction Decomposition Calculator }\)
\(\,\,\,\,\,\,\,\,\text{(Symbolab.com)}\)
\(\bullet\text{ Algebra 2/ Precalculus Homepage}\)
\(\,\,\,\,\,\,\,\,\text{All the Best Topics…}\)
\(\bullet\text{ Ratios and Proportions}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{4}{3}=\frac{d-4}{12}…\)
\(\bullet\text{ Rational Expressions- Multiplying and Dividing}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{x^2+3x-4}{(x+4)(x+5)}\cdot \displaystyle\frac{x+5}{x-1}…\)
\(\bullet\text{ Rational Expressions- Adding and Subtracting}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{x-5}{x+3}+\frac{x+2}{x^2+5x+6}…\)
\(\bullet\text{ Direct, Inverse, and Joint Variation}\)
\(\,\,\,\,\,\,\,\,\)
\(…\)
\(\bullet\text{ Complex Fractions}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{\frac{x}{5}+\frac{1}{3}}{\frac{1}{5}-\frac{1}{6}}…\)
\(\bullet\text{ Partial Fraction Decomposition}\)
\(\,\,\,\,\,\,\,\,\displaystyle\frac{8x+10}{x^2+2x}=\displaystyle\frac{5}{x} + \frac{3}{x+2}…\)
In Summary
Partial fraction decomposition, also known as partial fraction expansion, is an important mathematical technique used to express a fraction as the sum of simpler fractions. It is a useful tool for simplifying complex algebraic expressions and finding indefinite integrals. It is often taught in pre-calculus and calculus courses.
Partial fraction decomposition is the inverse operation for adding fractions together by finding a common denominator. Once this is understood, it is a little easier to see what is happening.
Topics related to Partial Fraction Decomposition
Rational Expressions: A rational expression is an expression that can be written as a fraction in which the numerator and denominator are both polynomials. Partial fraction decomposition is a technique that can be used to rewrite a rational expression as a sum of simpler fractions. It can make it easier to simplify a rational expression or to find the roots (solutions) of an equation involving a rational expression. It can also be helpful in finding the inverse of a rational function.
Indefinite Integrals: An indefinite integral is an antiderivative of a function. Partial fraction decomposition can be used to simplify indefinite integrals that involve rational functions.
Differential Equations: Partial fraction decomposition is often used in the solution of differential equations, particularly linear differential equations with constant coefficients. By expressing the solution as the sum of simpler fractions, it is easier to find the general solution to the differential equation.
Power Series: Partial fraction decomposition is used to find the power series representation of a function, which is a series expansion of the function in terms of a power of a variable. This can be used to approximate the function or to find its Taylor series expansion.
Complex Analysis: Partial fraction decomposition is used in complex analysis to express complex functions as the sum of simpler fractions. This can be used to find the Laurent series expansion of a function, which is a series expansion of the function around a singularity.
Matrix Algebra: Partial fraction decomposition is used in matrix algebra to express a matrix as the sum of simpler matrices. This can be used to find the inverse of a matrix or to decompose a matrix into simpler matrices using techniques such as the QR decomposition or the singular value decomposition.
Numerical analysis: Partial fraction decomposition is used in numerical analysis to approximate functions using polynomial functions. This can be used to find the least squares fit of a function to a set of data points or to interpolate a function using polynomial interpolation.