Rational functions - representation
Rational functions $f_{1}, f_{2}, f_{3}$ and $f_{4}$ are represented in two different ways:
$$ \begin{aligned}&f_{1}(x)=\frac{3 x+5}{x^{2}+2 x-3}=\frac{a}{x-1}+\frac{b}{x+3 }, \\&f_{2}(x)=\frac{2 x^{2}+5 x-1}{\left(x^{2}-1\right)(x+2)}=\frac {a}{x-1}+\frac{b}{x+1}+\frac{c}{x+2}, \\&f_{3}(x)=\frac{x^{2}+ 2 x-1}{x^{2}+x-2}=a+\frac{b}{x-1}+\frac{c}{x+2}, \\&f_{4}(x)= \frac{2 x^{2}+x-1}{x^{3}-x^{2}+x-1}=\frac{a}{x-1}+\frac{b x+c }{x^{2}+1} \end{aligned} $$
a) For each of the functions: determine the constants in the right-hand representation, so that the equality holds in the whole (implicit) definition set.
b) Also represent the following function as a sum of fractions:
$$f_{5}(x)=\frac{x+1}{(x-1)\left(x^{2}+4\right)}$$
a) For each of the functions : determine the constants in the right-hand representation, so that the equality holds in the whole (implicit) definition set.
My solution
For the function $f_1$ applies:
$$ \begin{aligned}&f_{1}(x)=\frac{3 x+5}{x^{2}+2 x-3}=\frac{a}{x -1}+\frac{b}{x+3} \\ \iff &3x+5 = a(x+3)+b(x-1) \\ \iff &a = \frac{3x+5}{x +3} \text{ and } b = \frac{3x-2}{x-1}\end{aligned} $$ For the function $f_2$ applies:
$$ \begin{aligned}&f_{2}(x) =\frac{2 x^{2}+5 x-1}{\left(x^{2}-1\right)(x+2)}=\frac{a}{x-1}+\frac {b}{x+1}+\frac{c}{x+2} \\ \iff &2x^2+5x-1 = a(x+2)+b(x-1)+c(x+1 ) \\ \iff &a = \frac{2x^2+5x-1}{x^2-1} \text{, } b = \frac{2x^2-4x-1}{x^2-1} \text{ and } c = \frac{-2x^2+9x-1}{x^2-1}\end{aligned} $$ For the function $f_3$ applies:
$$ \begin{aligned}&f_{3 }(x)=\frac{x^{2}+2 x-1}{x^{2}+x-2}=a+\frac{b}{x-1}+\frac{c}{x +2} \\ \iff &x^2+2x-1 = a(x^2+x-2)+b(x+2)+c(x-1) \\ \iff &a = \frac{x^ 2+2x-1}{x^2+x-2} \text{, } b = \frac{-3x^2+8x-1}{x^2+x-2} \text{ and } c = \frac{x^2-4x-1}{x^2+x-2} \end{aligned} $$
For the function $f_4$ applies:
$$ \begin{aligned}&f_{4}(x)=\ frac{2 x^{2}+x-1}{x^{3}-x^{2}+x-1}=\frac{a}{x-1}+\frac{b x+c} {x^{2}+1} \\ \iff &2x^2+x-1 = a(x^2-x+1)+b(x^2 +1)+(bx+c)(x-1) \\ \iff &a = \frac{2x^2+x-1}{x^2-x+1} \text{, } b = \frac{ -3x^2+x-1}{x^2-x+1} \text{ and } c = \frac{x^3-4x+1}{x^2-x+1} \end{aligned} $$
b) Also represent the following function as a sum of fractions:
My solution
$$ f_{5}(x)=\frac{x+1}{(x-1)\left(x^{2}+4\right)}$$
For the function $f_5$ applies:
$$ \begin{aligned}&f_{5}(x)=\frac{x+1}{(x-1)\left(x^{2}+4\right) }=\frac{a}{x-1}+\frac{b x+c}{x^{2}+4} \\ \iff &x+1 = a(x^2+4)+(bx+ c)(x-1) \\ \iff &a = \frac{x+1}{x^2+4} \text{, } b = \frac{-x-1}{x^2+4} \text{and} c = \frac{x^2-1}{x^2+4} \end{aligned} $$