What do you do first with an improper fraction?

Divide to get a polynomial plus a proper fraction (or include a constant term in the form).

Form of \frac{px^2+qx+r}{(x-a)(x-b)^2}?

\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{(x-b)^2}

How do you find A in the cover-up/substitution method?

Multiply by the denominator, then substitute the value of x that makes the other brackets zero.

After using substitution for a repeated factor, how do you find the last constant?

Compare coefficients (for example of x^2) or substitute another value such as x=0.

How do partial fractions help with a series expansion?

Expand each simple fraction with the binomial expansion, then add the results.

Are quadratic factors such as x^2+4 needed in this topic?

No. Only linear factors, including repeated ones.

Partial fractionsEdexcel International A Level Maths: Flashcards

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What is a proper fraction?

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What is a proper fraction?: One whose numerator has lower degree than its denominator.
What do you do first with an improper fraction?: Divide to get a polynomial plus a proper fraction (or include a constant term in the form).
Form of $\frac{px+q}{(x-a)(x-b)}$ in partial fractions?: $\frac{A}{x-a}+\frac{B}{x-b}$
Form of $\frac{px^2+qx+r}{(x-a)(x-b)^2}$ ?: $\frac{A}{x-a}+\frac{B}{x-b}+\frac{C}{(x-b)^2}$
How do you find A in the cover-up/substitution method?: Multiply by the denominator, then substitute the value of x that makes the other brackets zero.
After using substitution for a repeated factor, how do you find the last constant?: Compare coefficients (for example of $x^2$ ) or substitute another value such as $x=0$ .
$\int\frac{A}{ax+b}\,\mathrm{d}x$ ?: $\frac{A}{a}\ln|ax+b|+c$
$\int\frac{C}{(ax+b)^2}\,\mathrm{d}x$ ?: $-\frac{C}{a(ax+b)}+c$
Write $\frac{10x-1}{(x-1)(2x+1)}$ in partial fractions.: $\frac{3}{x-1}+\frac{4}{2x+1}$
Write $\frac{3x^2+6x-3}{(x-1)(x+2)}$ in partial fractions.: $3+\frac{2}{x-1}+\frac{1}{x+2}$
How do partial fractions help with a series expansion?: Expand each simple fraction with the binomial expansion, then add the results.
Are quadratic factors such as $x^2+4$ needed in this topic?: No. Only linear factors, including repeated ones.

Exam questions on Partial fractions

The function $f(x)=\frac{10x-1}{(x-1)(2x+1)}$ can be written in the form $\frac{A}{x-1}+\frac{B}{2x+1}$ .
Hence find $\int f(x)\,\mathrm{d}x$ .2 marks
The function $g(x)=\frac{3x^2+6}{(x+2)(x-1)^2}$ can be written in the form $\frac{A}{x+2}+\frac{B}{x-1}+\frac{C}{(x-1)^2}$ .
Find the value of $B$ .2 marks
The function $h(x)=\frac{3x^2+6x-3}{(x-1)(x+2)}$ is defined for $x>1$ .
Express $h(x)$ in the form $P+\frac{Q}{x-1}+\frac{R}{x+2}$ , where $P$ , $Q$ and $R$ are constants.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).