$$\DeclareMathOperator{cosec}{cosec}$$

# Number and Algebra

Syllabus Content

Partial fractions

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## Furthermore

Official Guidance, clarification and syllabus links:

Maximum of two distinct linear terms in the denominator, with degree of numerator less than the degree of the denominator.

Example

$$\dfrac{2x+1}{x^2+x-2} \equiv \dfrac{A}{(x-1)} + \dfrac{B}{(x+2)}$$

Here is an example of how to use partial fractions to decompose the rational function $$\dfrac{2x-5}{(x-2)(x+1)}$$:

\begin{aligned} \frac{2x-5}{(x-2)(x+1)} &= \frac{A}{x-2} + \frac{B}{x+1} \\ &= \frac{A(x+1) + B(x-2)}{(x-2)(x+1)} \end{aligned}

Multiplying both sides by the common denominator gives:

$$2x-5 = A(x+1) + B(x-2)$$

Substituting $$x = 2$$ gives:

$$-1 = 3A$$ $$A = -\dfrac{1}{3}$$

Substituting $$x = -1$$ gives:

$$-7 = -3B$$ $$B = \dfrac{7}{3}$$

Therefore, we have:

$$\frac{2x-5}{(x-2)(x+1)} = -\frac{1}{3(x-2)} + \frac{7}{3(x+1)}$$

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