Question 1.42: Express as partial fractions 4x³+10x+4/2x²+x...
Express as partial fractions
\frac{4 x^3+10 x+4}{2 x^2+x}The degree of the numerator is 3, that is n = 3. The degree of the denominator is 2, that is d = 2. Thus, the fraction is improper.
Now n — d = 1 and this is a measure ofthe extent to which the fraction is improper.
The partial fractions will include a polynomial of degree 1, that is Ax + B, in addition
to the partial fractions generated by the factors of the denominator.
The denominator factorizes to x{2x +1). These factors generate partial fractions of the form \frac{C}{x}+\frac{D}{2 x+1}. Hence
Multiplying by x and 2x + 1 yields
4 x^3+10 x+4=(A x+B) x(2 x+1)+C(2 x+1)+D x (1.12)
The constants A, B, C and D can now be evaluated.
Putting x = 0 into Equation (1.12) gives
4 = C
Putting x = —0.5 into Equation (1.12) gives
\begin{aligned} -1.5 & =-\frac{D}{2} \\D & =3\end{aligned}Equaling coefficients of x³ gives
4 = 2A
A = 2
Equaling coefficients of x gives
10= B + 2C + D
B = -1
Hence
\frac{4 x^3+10 x+4}{2 x^2+x}=2 x-1+\frac{4}{x}+\frac{3}{2 x+1}