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 an irreducible quadratic factor $x^2+bx+c$with multiplicity $m$, then for some constants $B_1,B_2,...,B_m$and $C_1,C_2,...,C_m$, 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 an irreducible quadratic factor $x^2+bx+c$with multiplicity $m$, then for some constants $B_1,B_2,...,B_m$ and $C_1,C_2,...,C_m$, the sum will include terms of the forms $\frac{B_{1}x+C_1}{\left(x^2+bx+c\right)}+\frac{B_{2}x+C_2}{{\left(x^2+bx+c\right)}^2}+...+\frac{B_{m}x+C_m}{{\left(x^2+bx+c\right)}^m}$.

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 an irreducible quadratic factor $x^2+bx+c$with multiplicity $m$, then for some constants $B_1,B_2,...,B_m$ and $C_1,C_2,...,C_m$, 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 an irreducible quadratic factor $x^2+bx+c$with multiplicity $m$, then for some constants $B_1,B_2,...,B_m$and $C_1,C_2,...,C_m$, the sum will include terms of the forms $\frac{B_{1}x+C_1}{\left(x^2+bx+c\right)}+\frac{B_{2}x+C_2}{{\left(x^2+bx+c\right)}^2}+...+\frac{B_{m}x+C_m}{{\left(x^2+bx+c\right)}^m}$.

Proper rational functions are decomposed into partial fractions for irreducible quadratic factor with some multiplicity as shown above.