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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 \(Q(x)\).

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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!

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 1/Q(x) in the form A/(x + 1) + B/(2x + 1), where A, B ∈ Z.
(d) Hence, or otherwise, show that 1/((x + 1)Q(x)) = 4/(2x + 1) - 2/(x + 1) - 1/(x + 1)^2.
(e) Hence, find the integral of 1/((x + 1)^2(2x + 1)) dx.
Consider the function defined by f(x) = P(x)/((x + 1)Q(x)), where x ≠ -1, x ≠ -1/2.
(f) Find the limits:
(i) lim (x -> -1) f(x);
(ii) lim (x -> ∞) f(x).

Step-by-step guide to solving polynomial factorization, partial fractions, and integration problems. Learn to apply the Factor Theorem, perform polynomial division, decompose fractions, and evaluate integrals and limits.

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