Resolve Rational Functions into Partial Fractions (Part I)
The last section on partial fractions concerns itself with the following: 2x−1 −1x+2=2(x+2)−(x−1)(x−1)(x+2)=x+5×2+x−2
On the left-hand side, we have an expression of one fraction minus another one. In order to combine them, we must first make a common denominator by multiplying the denominator of one fraction by the numerator of another. We then adjust the numerator accordingly and simplify it before adding or subtracting them together.
The final result will be a single fraction. If we integrate the expression on the right-hand side of this equation, it may look difficult to do so. However, if we know that this fraction is equal to the difference of these two simple fractions, then we can easily integrate the left-hand side. Partial fraction decomposition is used to reverse this process. Partial fraction decomposition is a technique used to find the sum of a number of simple fractions that equals a more complex one.
It can be thought of as the reverse of the process used to write down an expression as a sum of fractions, which is called partial fraction decomposition.
x+5×2+x−2=x+5(x−1)(x+2)Ax−1 +Bx+2
When adding or subtracting fractions, we can see that these denominators actually come from the same denominator by factorization. 2x−1 −1x+2=2(x+2)−(x−1)(x−1)(x+2)=x+5×2+x−2
When we are given a fraction such as this, we first look to factorize the denominators.
2x−1 −1x+2=2(x+2)−(x−1)(x−1)(x+2)=x+5×2+x−2
By factoring the denominator, we know how the functions on the left-hand side should look like. They should have denominators of these factors. We know that the fractions should be of this shape with denominators so we set the left-hand side to be this shape, with some constants A and B.
x−1 +Bx+2
To find the values of these two numbers, we first need to introduce some terminology for fractions in which the numerator and denominator are polynomials.
x+5×2+x−2,×2−x+5×2+x−2
A rational function is a fraction in which the numerator and denominator are both polynomials in x.
Here is an example:
x+5×2+x−2
This is an example of a rational function. The numerator is the polynomial x plus five, while the denominator is the polynomial x squared plus x minus two.
x2−x+5×2+x−2
A rational function is a quotient of two polynomials. It can be expressed in general form as f(x)/g(x), where f(x) and g(x) are both polynomials. a rational functionf(x)g(x)is proper if deg deg f < deg deg ge. g.x+5×2+x−2is proper but x2−x+5×2+x−2is not proper
A rational function is proper if the numerator has degree less than the degree of the denominator. In other words, if f is less than g. Here, this is an example of a proper rational function because the numerator is degree one and the denominator is degree two.
x+5×2+x−2
However, this example is not proper, because the numerator has degree two, which is equal to the degree of the denominator.
2−x+5×2+x−2
Every non-proper rational function can be transformed into a proper rational function.
2−x+5×2+x−2= 1 +−2x+7×2+x−2
If the degree of a rational function is too large, we can divide its numerator by its denominator. We do this by long division. The remainder is our new numerator and the original denominator becomes our new denominator. This makes it possible to transform a non-proper rational function into a polynomial together with a proper rational function.
2x−1 −1x+2=2(x+2)−(x−1)(x−1)(x+2)=x+5×2+x−2
x+5×2+x−2=x+5(x−1)(x+2)
Ax−1 +Bx+2
The form of the fractions on the left-hand side should match that of the given fraction when using partial fractions.
x−1 −1x+2=2(x+2)−(x−1)(x−1)(x+2)=x+5×2+x−2
The form of a fraction is the relationship between its numerator and denominator. The factors of the denominator come from the factorization of the numerator; therefore, it’s very important that we can factorize the right-hand side.
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