11

Consider the polynomial $P(x)=3x^{3}+5x^{2}+x-1$.

(a) Show that $(x+1)$ is a factor of $P(x)$.

(b) Hence, express $P(x)$ as a product of three linear factors.

Now consider the polynomial $Q(x)=(x+1)(2x+1)$.

(c) Express $\displaystyle\frac{1}{Q(x)}$ in the form $\dfrac{A}{x+1}+\dfrac{B}{2x+1}$, where $A,B\in\mathbb Z$.

(d) Hence, or otherwise, show that

$\frac{1}{(x+1)Q(x)}=\frac{4}{2x+1}-\frac{2}{x+1}-\frac{1}{(x+1)^{2}}.$

(e) Hence, find

$\int \frac{dx}{(x+1)^{2}(2x+1)}.$

Consider the function

$f(x)=\frac{P(x)}{(x+1)Q(x)},\qquad x\neq-1,\;x\neq-\tfrac12 .$

(f) Find

(i) $\displaystyle\lim_{x\to-1} f(x)$;
(ii) $\displaystyle\lim_{x\to\infty} f(x)$.