a) Express $f(x) = \frac{3 - x}{(1 - 2x)(1 + x^2)}$ in partial fractions:

First, let's assume that the function can be decomposed into partial fractions of the following form:

$\frac{3 - x}{(1 - 2x)(1 + x^2)} = \frac{A}{1 - 2x} + \frac{Bx + C}{1 + x^2}$

Here, $A$, $B$, and $C$ are constants that need to be determined.

Multiply both sides by the denominator $(1 - 2x)(1 + x^2)$:

$3 - x = A(1 + x^2) + (Bx + C)(1 - 2x)$

Expand both terms:

$3 - x = A(1 + x^2) + (Bx + C)(1 - 2x)$ $3 - x = A + Ax^2 + Bx - 2Bx^2 + C - 2Cx$ $3 - x = (A + C) + (B - 2C)x + (A - 2B)x^2$

Now, compare the coefficients of $1$, $x$, and $x^2$ on both sides:

1. Constant term: $A + C = 3$
2. Coefficient of $x$: $B - 2C = -1$
3. Coefficient of $x^2$: $A - 2B = 0$

Solve this system of equations:

• From $A - 2B = 0$, we get $A = 2B$.
• Substitute $A = 2B$ into $A + C = 3$: $2B + C = 3 \quad \text{(1)}$
• From $B - 2C = -1$, we have: $B = 2C - 1 \quad \text{(2)}$

Substitute equation (2) into equation (1):

$2(2C - 1) + C = 3$ $4C - 2 + C = 3$ $5C = 5$ $C = 1$

Now substitute $C = 1$ into equation (2):

$B = 2(1) - 1 = 1$

Substitute $B = 1$ into $A = 2B$:

$A = 2(1) = 2$

Thus, the partial fraction decomposition is:

$\frac{3 - x}{(1 - 2x)(1 + x^2)} = \frac{2}{1 - 2x} + \frac{x + 1}{1 + x^2}$

b) Obtain the expansion of $f(x)$ in ascending powers of $x$:

Now, expand each term separately in ascending powers of $x$.

1. Expand $\frac{2}{1 - 2x}$ using the binomial series: $\frac{2}{1 - 2x} = 2(1 - 2x)^{-1} = 2\left(1 + 2x + (2x)^2 + \cdots\right) = 2\left(1 + 2x + 4x^2 + \cdots\right)$ Up to the term in $x^2$, this becomes: $\frac{2}{1 - 2x} = 2 + 4x + 8x^2$

2. Expand $\frac{x + 1}{1 + x^2}$: $\frac{x + 1}{1 + x^2} = (x + 1)\left(1 - x^2 + x^4 - \cdots\right)$ Multiply $(x + 1)$ by the series: $(x + 1)(1 - x^2) = x + 1 - x^2 - x^3$ Up to the term in $x^2$, this becomes: $\frac{x + 1}{1 + x^2} = x + 1 - x^2$

3. Combine the two expansions: $f(x) = \left(2 + 4x + 8x^2\right) + \left(x + 1 - x^2\right)$ Simplify: $f(x) = (2 + 1) + (4x + x) + (8x^2 - x^2)$ $f(x) = 3 + 5x + 7x^2$

Thus, the expansion of $f(x)$ up to the term in $x^2$ is:

$f(x) = 3 + 5x + 7x^2$

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Would you like more details or clarification on any part of this solution?

Related Questions:

1. How does the binomial expansion work for terms like $(1 - 2x)^{-1}$?
2. Can partial fraction decomposition always be used for rational functions?
3. How would the expansion differ if higher powers of $x$ were included?
4. What happens to the expansion if $f(x)$ has higher-degree polynomials in the denominator?
5. How can you check if the partial fraction decomposition is correct?

Tip:

When expanding functions using series, it helps to know the radius of convergence to understand where the expansion is valid.