I don't understand partial fraction decomposition

In summary, when dealing with partial fraction problems, if there is a term in the denominator that is in the form of (x^2+3x+6), we use the form Ax+B rather than just A. Similarly, if the denominator is (x^2+3x+6)^2, we use the form {(Ax+B)/(x^2+3x+6)}+{(Cx+D)/(x^2+3x+6)^2}. This setup may seem arbitrary, but it is derived from considering simpler cases, such as when the initial fraction is (4x-2)/(x^2+3x+6), which cannot be written in the form A/(x^2+3x
  • #1
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if there is something like (x^2+3x+6) in the denominator for one of the terms in a partial fraction problem, why do we put Ax+B instead of just A? and if the denominator is (x^2+3x+6)^2, why do we do {(Ax+B)/(x^2+3x+6)}+{(Cx+D)/(x^2+3x+6)^2}? i was always just told to memorize it, but why do we set it up this way? even a link to a derivation will probably help. the only thing i can find online is the formulas, basically what i just wrote.. but i want to know why those setups are the way they are, ie the reason behind them.
 
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  • #2
Have you thought about the simplest cases? Suppose your initial fraction is just [itex](4x- 2)/(x^2+ 3x+ 6)[/itex]. Do you think you ought to be able to write that as [itex]A/(x^2+ 3x+ 6)[/itex]? What would have happened to the "4x"?
 

1. What is partial fraction decomposition?

Partial fraction decomposition is a mathematical process used to break down a rational expression into simpler fractions. It is often used in calculus to solve integrals and simplify algebraic expressions.

2. Why is partial fraction decomposition important?

Partial fraction decomposition allows us to simplify complex rational expressions and make them easier to work with. It also helps us solve integrals and evaluate limits in calculus.

3. How do you perform partial fraction decomposition?

To perform partial fraction decomposition, we first factor the denominator of the rational expression. Then, we set up a system of equations using the coefficients of the factors in the denominator. Finally, we solve the system of equations to determine the unknown coefficients of the simpler fractions.

4. What are the different types of partial fraction decomposition?

There are two main types of partial fraction decomposition: proper and improper. Proper decomposition involves breaking down a rational expression with a proper fraction in the numerator. Improper decomposition is used when the degree of the numerator is greater than or equal to the degree of the denominator.

5. How is partial fraction decomposition used in real life?

Partial fraction decomposition is used in a variety of fields, including engineering, physics, and economics. It is used to solve differential equations, simplify circuit analysis, and model real-world situations. It is also used in signal processing to analyze and manipulate signals.

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