Introduction:
If f(x) and g(x) are two polynomials, then $\frac{f(x)} {g(x)}$ defines a rational algebraic function or a rational function of x.
If degree of f(x) < degree of of g (x), then $\frac{f(x)} {g(x)}$ is called a proper rational function.
If degree of f (x) > degree of g(x) then $\frac{f(x)} {g(x)}$ is called an improper rational function.
If $\frac{f(x)} {g(x)}$is an improper rational function, we divide f(x) by g(x) so that the rational function $\frac{f(x)} {g(x)}$ is expressed in the form F (x) + ?$\frac{f(x)} {g(x)}$ where F(x)and ?(x) are polynomials such that the degree of ?(x)is less than that of g(x)is less than that f(x). Thus, $\frac{f(x)} {g(x)}$ is expressible as the sum of a polynomial and a proper rational functions.
Any proper rational function $\frac{f(x)} {g(x)}$ can be expressed as the sum of rational functions, each having a simple factor of g(x). Each such fraction is called a partial fraction and the process of obtaining then is called the resolution or decomposition or decomposition of $\frac{f(x)} {g(x)}$ into partial fractions.
How to Find Partial Fraction:
The resolution of $\frac{f(x)} {g(x)}$into partial fractions depends mainly upon the nature of the factors of g(x) as discussed below.
Case:- When denominator is expressible as the product of non-repeating linear factors.I like to share this Cdf of Uniform Distribution with you all through my article.
Let g(x) = (x – a1) (x – a2) … (x – an). Then we assume that
$\frac{f(x)} {g(x)}$ = A1/ x + A2/x x – a2 + … + An/x – an
where A1, A2, … An are constants and can be determined by equating the numerator on RHS to the numerator o LHS and then substituting x = a1, a2, …, an.
My Previous Blog :- http://wanttolearnmath.blogspot.in/2012/11/multiplying-three-factors.html
If f(x) and g(x) are two polynomials, then $\frac{f(x)} {g(x)}$ defines a rational algebraic function or a rational function of x.
If degree of f(x) < degree of of g (x), then $\frac{f(x)} {g(x)}$ is called a proper rational function.
If degree of f (x) > degree of g(x) then $\frac{f(x)} {g(x)}$ is called an improper rational function.
If $\frac{f(x)} {g(x)}$is an improper rational function, we divide f(x) by g(x) so that the rational function $\frac{f(x)} {g(x)}$ is expressed in the form F (x) + ?$\frac{f(x)} {g(x)}$ where F(x)and ?(x) are polynomials such that the degree of ?(x)is less than that of g(x)is less than that f(x). Thus, $\frac{f(x)} {g(x)}$ is expressible as the sum of a polynomial and a proper rational functions.
Any proper rational function $\frac{f(x)} {g(x)}$ can be expressed as the sum of rational functions, each having a simple factor of g(x). Each such fraction is called a partial fraction and the process of obtaining then is called the resolution or decomposition or decomposition of $\frac{f(x)} {g(x)}$ into partial fractions.
How to Find Partial Fraction:
The resolution of $\frac{f(x)} {g(x)}$into partial fractions depends mainly upon the nature of the factors of g(x) as discussed below.
Case:- When denominator is expressible as the product of non-repeating linear factors.I like to share this Cdf of Uniform Distribution with you all through my article.
Let g(x) = (x – a1) (x – a2) … (x – an). Then we assume that
$\frac{f(x)} {g(x)}$ = A1/ x + A2/x x – a2 + … + An/x – an
where A1, A2, … An are constants and can be determined by equating the numerator on RHS to the numerator o LHS and then substituting x = a1, a2, …, an.
My Previous Blog :- http://wanttolearnmath.blogspot.in/2012/11/multiplying-three-factors.html
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