Question

Fill in the blanks to complete each of the following theorem statements:

In a partial-fractions decomposition of a proper rational function $\frac{p\left(x\right)}{q\left(x\right)}$, if $q\left(x\right)$has a linear factor $x-c$with multiplicity $m$, then for some constants $A_1,A_2,...,A_m$, the sum will include terms of the forms _____.

Short answer

In a partial-fractions decomposition of a proper rational function $\frac{p\left(x\right)}{q\left(x\right)}$, if $q\left(x\right)$has a linear factor $x-c$with multiplicity $m$, then for some constants $A_1,A_2,...,A_m$, the sum will include terms of the forms$\frac{A_1}{\left(x-c\right)}+\frac{A_2}{{\left(x-c\right)}^2}+...+\frac{A_m}{{\left(x-c\right)}^m}$.

Step-by-step solution

Step 1. Given information  

In a partial-fractions decomposition of a proper rational function $\frac{p\left(x\right)}{q\left(x\right)}$, if $q\left(x\right)$has a linear factor $x-c$with multiplicity $m$, then for some constants $A_1,A_2,...,A_m$, the sum will include terms of the forms _____.

Step 2. Filling in the blanks to complete the theorem statements  

In a partial-fractions decomposition of a proper rational function $\frac{p\left(x\right)}{q\left(x\right)}$, if $q\left(x\right)$has a linear factor $x-c$with multiplicity $m$, then for some constants $A_1,A_2,...,A_m$, the sum will include terms of the forms $\frac{A_1}{\left(x-c\right)}+\frac{A_2}{{\left(x-c\right)}^2}+...+\frac{A_m}{{\left(x-c\right)}^m}$.

Proper rational functions are decomposed into partial fractions for linear factors with some multiplicity as shown above.