Let's break down the question step by step. Here's how to approach each part of the problem:

(a) Show that $(x + 1)$ is a factor of $P(x) = 3x^3 + 5x^2 + x - 1$.

To prove $(x + 1)$ is a factor of $P(x)$, we use the Factor Theorem. According to this theorem, if $x + 1$ is a factor of $P(x)$, then substituting $x = -1$ into $P(x)$ should give 0.

Let's substitute $x = -1$ into $P(x)$:

$P(-1) = 3(-1)^3 + 5(-1)^2 + (-1) - 1 = -3 + 5 - 1 - 1 = 0$

Since $P(-1) = 0$, $(x + 1)$ is a factor of $P(x)$.

(b) Express $P(x)$ as a product of three linear factors.

Now that we know $(x + 1)$ is a factor, we can perform synthetic or polynomial division of $P(x)$ by $x + 1$ to find the remaining quadratic factor.

Performing division of $P(x) = 3x^3 + 5x^2 + x - 1$ by $x + 1$ will give the quotient, which is a quadratic expression.

Let's perform the division:

• Dividing $3x^3$ by $x$, we get $3x^2$.
• Multiply $3x^2(x + 1)$, subtract from the original polynomial, and continue this process.

Once the quotient is found, factorize it to find the three linear factors.

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

Here, $Q(x) = (x + 1)(2x + 1)$. We want to express:

$\frac{1}{(x + 1)(2x + 1)} = \frac{A}{x+1} + \frac{B}{2x+1}$

To solve for $A$ and $B$, multiply both sides of the equation by $(x+1)(2x+1)$ and solve the resulting equation by equating coefficients of powers of $x$.

(d) Hence, or otherwise, show that:

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

Start with the expression from part (c), then manipulate it algebraically to match the given expression.

(e) Hence, find $\int \frac{1}{(x+1)^2(2x+1)} \, dx$.

Use the result from part (d) to decompose the integrand into simpler fractions that can be integrated term by term.

(f) Find the following limits for the function $f(x) = \frac{P(x)}{(x+1)Q(x)}$:

(i) $\lim_{x \to -1} f(x)$

(ii) $\lim_{x \to \infty} f(x)$

For part (i), analyze the behavior of $f(x)$ as $x \to -1$ to check for continuity or potential indeterminate forms like $\frac{0}{0}$.

For part (ii), compute the limit as $x$ approaches infinity using the highest degree terms in $P(x)$ and $Q(x)$.

Would you like detailed steps for any specific part of this question? Here are five related problems to deepen understanding:

1. Perform synthetic division of $P(x) = 3x^3 + 5x^2 + x - 1$ by $x + 1$.
2. Find the quadratic factor of $P(x)$ after division by $x + 1$.
3. Verify the values of $A$ and $B$ in part (c) for a different quadratic $Q(x)$.
4. Solve the integral $\int \frac{1}{(x+1)(2x+1)} \, dx$.
5. Calculate the limit $\lim_{x \to -\frac{1}{2}} \frac{P(x)}{(x+1)Q(x)}$.

Tip: Always verify the form of the integrand before integrating—it may simplify calculations!