Partial Fractions (Cambridge International A-Level Mathematics): Exam Questions

Express

\frac{1}{x \left(x + 1\right)}

as partial fractions.

(i) Factorise $x^{2} + 5 x - 6$.

(ii) Hence, or otherwise, express

$\frac{5x+16}{x^{2}+5x-6}$

as partial fractions.

Express

$\frac{6x-13}{x^{2}-5x+6}$

as partial fractions.

Let

\text{f}(x) = \frac{2 \left(x - 11\right)}{x^{2} + 2 x - 15}

Express $\text{f} ( x )$ in partial fractions.

Let

\text{f}(x) = \frac{3 x + 8}{\left(x + 1\right)^{2}}

Express $\text{f} ( x )$ in partial fractions.

Let

\text{f}(x) = \frac{2}{x \left(x + 2\right)}

Express $\text{f} ( x )$ in partial fractions.

Let

\text{f}(x) = \frac{x - 10}{\left(x + 2\right) \left(x - 1\right)}

Express $\text{f} ( x )$ in partial fractions.

Express $\frac{7x+34}{x^{2}+9x+14}$ as partial fractions.

Express \frac{2 x}{\left(x + 1\right) \left(x - 2\right)} as partial fractions.

Let

$\text{f} ( x ) = \frac{12}{x^{2}+4x-5}$

Express $\text{f} ( x )$ in partial fractions.

Let

\text{f}(x) = \frac{12 x - 6}{\left(x - 3\right) \left(x + 2\right) \left(x - 1\right)}

Express $\text{f} ( x )$ in partial fractions.

Let

$\text{f} ( x ) = \frac{2x}{x^{2}-9x-52}$

Express $\text{f} ( x )$ in partial fractions.

Let

\text{f}(x) = \frac{5 x^{2} + 10 x + 8}{x \left(x + 2\right)^{2}}

Express $\text{f} ( x )$ in partial fractions.

Let

$\text{f} ( x ) = \frac{3x^{2}-6x+2}{x^{3}-3x^{2}+2x}$

Express $\text{f} ( x )$ in partial fractions.

(i)

Briefly explain why

$\frac{x^{2}+x-4}{x^{2}+4x+3}$

is an improper (algebraic) fraction.

(ii)

Show that

\left(x^{2} + x - 4\right) \div \left(x^{2} + 4 x + 3\right) = 1 - \frac{\left(3 x + 7\right)}{x^{2} + 4 x + 3}

Express

$\frac{3x+7}{x^{2}+4x+3}$

as partial fractions.

Hence, show that

$\frac{x^{2}+x-4}{x^{2}+4x+3} = 1 - \frac{2}{x+1} - \frac{1}{x+3}$

Let

\text{f}(x) = \frac{x^{2} + x + 5}{\left(x + 2\right) \left(3 + x^{2}\right)}

Express $\text{f} ( x )$ in partial fractions.

Let

\text{f}(x) = \frac{3 \left(x^{2} - x + 1\right)}{\left(x + 1\right) \left(x - 2\right)^{2}}

Express $\text{f} ( x )$ in partial fractions.

It is given that

\text{f} \left(x\right) = x^{3} - 13 x + 12 .

(i) Show that \text{f} \left(1\right) = 0.

(ii) Hence write down a factor of \text{f} \left(x\right).

(i) Show that \text{f} \left(3\right) = 0 .

(ii) Hence write down another factor of \text{f} \left(x\right).

Fully factorise \text{f} \left(x\right).

Express

$\frac{3x^{2}-13}{x^{3}-13x+12}$

in partial fractions.

Use algebraic division to work out \left(x^{2} + 4 x - 6\right) \div \left(x^{2} - x - 6\right).

Express \frac{5 x}{\left(x + 2\right) \left(x - 3\right)} in partial fractions.

Hence, or otherwise, show that $\frac{x^{2}+4x-6}{x^{2}-x-6} = 1 + \frac{2}{x+2} + \frac{3}{x-3}$

Use the factor theorem to show that \left(x + 1\right) is a factor of $x^{3} - 3 x^{2} + 4$.

Hence, or otherwise, fully factorise $x^{3} - 3 x^{2} + 4$.

Express

$\frac{19-8x}{x^{3}-3x^{2}+4}$

in partial fractions.

Let

$\text{f} ( x ) = \frac{3x+11}{x^{2}+6x+9}$

Express $\text{f} ( x )$ in partial fractions.

Let

\text{f}(x) = \frac{11 x^{2} - 2 x + 14}{\left(2 x - 1\right) \left(5 + x^{2}\right)}

Express $\text{f} ( x )$ in partial fractions.

Let

\text{f}(x) = \frac{ 4 - 4 x - 4 x^{2} }{\left(3 x - 1\right) \left(x^{2} - 2 x + 1\right)}

Express $\text{f} ( x )$ in partial fractions.

It is given that

\text{f} \left(x\right) = x^{3} + 3 x^{2} - 4 x - 12 .

(i) Show that \text{f} \left(- 3\right) = 0

(ii) Hence write down a factor of \text{f} \left(x\right).

Fully factorise \text{f} \left(x\right).

Express

$\frac{3x^{2}-8x-36}{x^{3}+3x^{2}-4x-12}$

in partial fractions.

Let

$\text{f} ( x ) = \frac{2x^{2}+16x-24}{x^{3}-2x^{2}-8x}$

Express $\text{f} ( x )$ in partial fractions.

It is given that

$\text{f} ( x ) = \frac{x^{2}+x+5}{x^{2}+3x+2}$

Show that

\text{f}(x) \equiv 1 - \frac{2 x - 3}{\left(x + 1\right) \left(x + 2\right)}

Express \frac{2 x - 3}{\left(x + 1\right) \left(x + 2\right)} in partial fractions.

Hence express $\text{f} ( x )$ in partial fractions.

Use the factor theorem to show that \left(x - 2\right) is a factor of $x^{3} + 4 x^{2} - 3 x - 18$.

Hence, or otherwise, fully factorise $x^{3} + 4 x^{2} - 3 x - 18$.

Express

$\frac{7x^{2}+32x+8}{x^{3}+4x^{2}-3x-18}$

in partial fractions.

Let

$\text{f} ( x ) = \frac{8}{x^{3}-6x^{2}+8x}$

Express $\text{f} ( x )$ in partial fractions.

Let

$\text{f} ( x ) = \frac{13x^{2}-10}{x^{3}-x^{2}-2x}$

Express $\text{f} ( x )$ in partial fractions.

Let

\text{f}(x) = \frac{x^{2} - 3}{\left(x - 3\right) \left(x + 2\right)^{2}}

Express $\text{f} ( x )$ in partial fractions.

Let

$\text{f} ( x ) = \frac{x^{2}-5}{x^{3}-x^{2}-17x-15}$

Express $\text{f} ( x )$ in partial fractions.

Let

$\text{f} ( x ) = \frac{3x^{2}-22x+25}{x^{3}-7x^{2}+8x+16}$

Express $\text{f} ( x )$ in partial fractions.

Let

$\text{f} ( x ) = \frac{2x^{2}-3}{x^{3}+4x^{2}-3x-18}$

Express $\text{f} ( x )$ in partial fractions.

Let

$\text{f} ( x ) = \frac{16x^{2}-7x+1}{3x^{3}-x^{2}}$

Express $\text{f} ( x )$ in partial fractions.

Given that \text{f} \left(\frac{1}{4}\right) = 0 , fully factorise \text{f} \left(x\right) where \text{f} \left(x\right) = 8 x^{3} - 2 x^{2} - 4 x + 1 .

Hence express \frac{5 x^{2} - 2 x - 2}{\text{f} \left(x\right)} in partial fractions.

Let

$\text{f} ( x ) = \frac{34-3x^{2}-x}{x^{3}-5x^{2}+3x+9}$

Express $\text{f} ( x )$ in partial fractions.

Let

$\text{f} ( x ) = \frac{x^{2}+3x+10}{x^{2}+8x+15}$

Express $\text{f} ( x )$ in partial fractions.

Let

$\text{f} ( x ) = \frac{2x-5}{x^{3}-3x^{2}-13x+15}$

Express $\text{f} ( x )$ in partial fractions.