Let \(f(x) = \frac{17x^2 - 7x + 16}{(2 + 3x^2)(2 - x)}\).

(a) Express \(f(x)\) in partial fractions.

(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^3\).

(c) State the set of values of \(x\) for which the expansion in (b) is valid. Give your answer in an exact form.

Let \(f(x) = \frac{16 - 17x}{(2 + x)(3 - x)^2}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{12 + 12x - 4x^2}{(2+x)(3-2x)}\).

(i) Express \(f(x)\) in partial fractions. [5]

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

Let \(f(x) = \frac{7x^2 - 15x + 8}{(1 - 2x)(2 - x)^2}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{x - 4x^2}{(3-x)(2+x^2)}\).

(i) Express \(f(x)\) in the form \(\frac{A}{3-x} + \frac{Bx+C}{2+x^2}\).

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^3\).

Let \(f(x) = \frac{12x^2 + 4x - 1}{(x-1)(3x+2)}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{8x^2 + 9x + 8}{(1-x)(2x+3)^2}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{5x^2 - 7x + 4}{(3x + 2)(x^2 + 5)}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{x(6-x)}{(2+x)(4+x^2)}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{3x^2 + x + 6}{(x+2)(x^2+4)}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{10x - 2x^2}{(x+3)(x-1)^2}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{24x + 13}{(1 - 2x)(2 + x)^2}\).

(a) Express \(f(x)\) in partial fractions.

(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

(c) State the set of values of \(x\) for which the expansion in (b) is valid.

Let \(f(x) = \frac{4x^2 + 12}{(x+1)(x-3)^2}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{5x^2 + x + 6}{(3 - 2x)(x^2 + 4)}\).

(i) Express \(f(x)\) in partial fractions. [5]

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

Let \(f(x) = \frac{x^2 - 8x + 9}{(1-x)(2-x)^2}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

(i) Express \(\frac{4 + 12x + x^2}{(3-x)(1+2x)^2}\) in partial fractions.

(ii) Hence obtain the expansion of \(\frac{4 + 12x + x^2}{(3-x)(1+2x)^2}\) in ascending powers of \(x\), up to and including the term in \(x^2\).

(i) Express \(\frac{7x^2 + 8}{(1+x)^2(2-3x)}\) in partial fractions.

(ii) Hence expand \(\frac{7x^2 + 8}{(1+x)^2(2-3x)}\) in ascending powers of \(x\) up to and including the term in \(x^2\), simplifying the coefficients.

Let \(f(x) = \frac{2x^2 - 7x - 1}{(x-2)(x^2+3)}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

(i) Express \(\frac{9 - 7x + 8x^2}{(3-x)(1+x^2)}\) in partial fractions.

(ii) Hence obtain the expansion of \(\frac{9 - 7x + 8x^2}{(3-x)(1+x^2)}\) in ascending powers of \(x\), up to and including the term in \(x^3\).

(i) Express \(\frac{5x - x^2}{(1+x)(2+x^2)}\) in partial fractions.

(ii) Hence obtain the expansion of \(\frac{5x - x^2}{(1+x)(2+x^2)}\) in ascending powers of \(x\), up to and including the term in \(x^3\).

Let \(f(x) = \frac{3x}{(1+x)(1+2x^2)}\).

(i) Express \(f(x)\) in partial fractions. [5]

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^3\). [5]

(i) Express \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in partial fractions.

(ii) Hence obtain the expansion of \(\frac{4 + 5x - x^2}{(1 - 2x)(2 + x)^2}\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{21 - 8x - 2x^2}{(1 + 2x)(3 - x)^2}\).

(a) Express \(f(x)\) in partial fractions.

(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

(i) Express \(\frac{1+x}{(1-x)(2+x^2)}\) in partial fractions.

(ii) Hence obtain the expansion of \(\frac{1+x}{(1-x)(2+x^2)}\) in ascending powers of \(x\), up to and including the term in \(x^2\).

(i) Express \(\frac{5x + 3}{(x + 1)^2(3x + 2)}\) in partial fractions.

(ii) Hence obtain the expansion of \(\frac{5x + 3}{(x + 1)^2(3x + 2)}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.

(i) Express \(\frac{2 - x + 8x^2}{(1-x)(1+2x)(2+x)}\) in partial fractions.

(ii) Hence obtain the expansion of \(\frac{2 - x + 8x^2}{(1-x)(1+2x)(2+x)}\) in ascending powers of \(x\), up to and including the term in \(x^2\).

(i) Express \(\frac{10}{(2-x)(1+x^2)}\) in partial fractions.

(ii) Hence, given that \(|x| < 1\), obtain the expansion of \(\frac{10}{(2-x)(1+x^2)}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients.

(i) Express \(\frac{3x^2 + x}{(x+2)(x^2+1)}\) in partial fractions.

(ii) Hence obtain the expansion of \(\frac{3x^2 + x}{(x+2)(x^2+1)}\) in ascending powers of \(x\), up to and including the term in \(x^3\).

Let \(f(x) = \frac{x^2 + 7x - 6}{(x-1)(x-2)(x+1)}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Show that, when \(x\) is sufficiently small for \(x^4\) and higher powers to be neglected,

\(f(x) = -3 + 2x - \frac{3}{2}x^2 + \frac{11}{4}x^3\).

Let \(f(x) = \frac{9x^2 + 4}{(2x + 1)(x - 2)^2}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Show that, when \(x\) is sufficiently small for \(x^3\) and higher powers to be neglected, \(f(x) = 1 - x + 5x^2\).

Let \(f(x) = \frac{6 + 7x}{(2-x)(1+x^2)}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Show that, when \(x\) is sufficiently small for \(x^4\) and higher powers to be neglected,

\(f(x) = 3 + 5x - \frac{1}{2}x^2 - \frac{15}{4}x^3\).

Let \(f(x) = \frac{2x^2 + 7x + 8}{(1+x)(2+x)^2}\).

(a) Express \(f(x)\) in partial fractions.

(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{5x^2 + 8x - 3}{(x-2)(2x^2 + 3)}\).

(a) Express \(f(x)\) in partial fractions.

(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{14 - 3x + 2x^2}{(2 + x)(3 + x^2)}\).

(a) Express \(f(x)\) in partial fractions.

(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{2 + 11x - 10x^2}{(1 + 2x)(1 - 2x)(2 + x)}\).

(a) Express \(f(x)\) in partial fractions. [5]

(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\). [5]

Let \(f(x) = \frac{8 + 5x + 12x^2}{(1-x)(2+3x)^2}\).

(a) Express \(f(x)\) in partial fractions.

(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).

Let \(f(x) = \frac{2x(5-x)}{(3+x)(1-x)^2}\).

(i) Express \(f(x)\) in partial fractions.

(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\).