Page 25 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes
Did u know? By substraction of a suitable multiple of an suitable inverse power, we can
attain a rational function that is less singular than R(x) / Q(x) at an random root of Q

Notes If we persist such substraction until we have detached all the singularities of R(x) /
Q(x) at all the roots, we will be left with a rational function that still disappears at infinity,
and now contains no finite singularities. The only such function is 0; so that R(x) / Q(x)
must equal the sum of the substractions; which statement is the theorem.

Self Assessment

Fill in the blanks:
1. By the method of “................................” we can translate any rational function into a
polynomial and fractions each one with negative powers of just one factor (x–a).
2. A ................................ function is defined as the proportion of two polynomials in the form
of P(x)/Q(x) , where P (x) and Q(x) are polynomials in x and Q( x) 0.
3. If the degree of P( x) is less than the degree of Q( x), then the rational function is known as
................................. .
4. The improper rational functions can be abridged to the proper rational functions by
................................ procedure.
5. Each polynomial can be factored into ................................ factors.
6. The degree of the numerator must be ................................ than the degree of the denominator.
7. The partial fraction theorem permits us to integrate P(X) / Q(x); we are required only to integrate
the polynomial D(x) and the different ................................ powers appearing in this sum.

2.2 Finding the Coefficients in the Partial Fraction Expansion

Here, we will find the a in the expression:
jk

)

R ( ) a  a ji (x q  a ( j m  1) (x q j ) m j  1
x
0 j
j
(*)   j
Q ( ) j (x q j ) m j
x

There are four different methods to do this.
2.2.1 Method 1: Expansion
Q ( )
x
x
Z ( ) 
Consider k m j
(x q j )

R ( )
x
Use the constant, linear, quadratic or higher approximation to at x  q j , to attain:
x
Z k ( )
1 d k R ( )
x
a  at x  q j .
jk
k
! k dx Z j ( )
x
constant, linear, quadratic or higher approximation
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