91Ó°ÊÓ

We need to write the summary for the Section $5.3$.

Suppose $\frac{p\left(x\right)}{q\left(x\right)}$is an improper rational function where $p\left(x\right)$and $q\left(x\right)$are polynomial functions with $deg\left(p\left(x\right)\right)=n$and $deg\left(q\right(x\left)\right)=m<n$. Then is an improper rational function where $p\left(x\right)$and $q\left(x\right)$are polynomial functions with $deg(p(x\left)\right)=n$and $deg\left(q\right(x\left)\right)=m<n$. Then

$\frac{p\left(x\right)}{q\left(x\right)}=s\left(x\right)+\frac{r\left(x\right)}{q{\left(x\right)}^{\text{'}}}$

for some polynomial $s\left(x\right)$of degree $n−m$and some proper rational function $\frac{r\left(x\right)}{q\left(x\right)}$with $deg\left(r\right(x\left)\right)<m.$

Every proper rational function $\frac{p\left(x\right)}{q\left(x\right)}$can be written as a sum of partial fractions, as follows:

1. If $q\left(x\right)$has a linear factor $x-c$with multiplicity $m$, then for some constants localid="1649648855901" $A_1,A_2,...,A_m$the sum will include terms of the form

localid="1649648861430" $\frac{A_1}{x-c}+\frac{A_2}{{\left(x-c\right)}^2}+....+\frac{A_m}{{\left(x-c\right)}^m}$

2. If $q\left(x\right)$has an irreducible quadratic factor localid="1649648865401" $x^2+bx+c$with multiplicity $m$, then for some constants localid="1649648869500" $B_1,B_2,....,B_m$and localid="1649648873958" $C_1,C_2,....,C_m$,the sum will include terms of the form.

localid="1649648878729" $\frac{B_{1}x+C_1}{x^2+bx+c}+\frac{B_{2}x+C_2}{{\left(x^2+bx+c\right)}^2}+.....+\frac{B_{m}x+C_m}{{\left(x^2+bx+c\right)}^m}$

$p\left(x\right)=q\left(x\right)m\left(x\right)+R\left(x\right)$

where $p\left(x\right),q\left(x\right)$are the two polynomials and $R\left(x\right)$is the remainder of the polynomial obtained by diving the polynomials and$m\left(x\right)$is the quotient polynomial.

In other words,

$\frac{p\left(x\right)}{q\left(x\right)}=m\left(x\right)+\frac{R\left(x\right)}{q\left(x\right)}$

The condition for Completing the Square:

Every quadratic function $x^2+bx+c$can be rewritten in the form

$x^2+bx+c={\left(x-k\right)}^2+C$

where $k=-\frac{b}{2},C=c-\frac{b^2}{4}$.