An algebraic fraction can be expressed in the form \( \frac{\mathrm{f}(x)}{\mathrm{g}(x)} \), where \(\mathrm{f}(x)\) and \(\mathrm{g}(x)\) are both polynomials. If \( \mathrm{f}(x)\) has a degree \(<\) \(\mathrm{g}(x)\), it is a proper algebraic fraction If \(\mathrm{f}(x)\) has a degree \( \geq\) \(\mathrm{g}(x)\), it is an improper algebraic fraction Proper algebraic fractions can be expressed as partial fractions depending on the denominator. The expression of partial fractions are summarised in the table: Denominator Algebraic Fraction Partial Fraction Non-Repeating Linear Factors \( \frac{px + q}{(ax + b)(cx + d)} \) \( \frac{A}{ax + b} + \frac{B}{cx + d}\) Repeating Linear Factors \( \frac{px + q}{(ax + b)^2(cx + d)} \) \(\frac{A}{ax + b} + \frac{B}{(ax + b)^2} + \frac{C}{cx + d} \) Quadratic Factors that cannot be factorised \( \frac{px + q}{(ax^2 + b)(cx + d)} \) \( \frac{Ax + B}{ax^2 + b} + \frac{C}{cx + d} \) The constants \(A, B, C,\) etc. can be found by equating the algebraic fraction to its partial fraction decomposition, clearing the denominators, and then comparing the coefficients of corresponding powers of \(x\). When dealing with improper algebraic fractions, perform long division on the fraction \( \frac{\mathrm{f}(x)}{\mathrm{g}(x)} \) first, so that: \( \frac{\mathrm{f}(x)}{\mathrm{g}(x)} = \mathrm{Q}(x) + \frac{\mathrm{R}(x)}{\mathrm{g}(x)}\) The fraction \( \frac{\mathrm{R}(x)}{\mathrm{g}(x)}\) is now a proper algebraic fraction. It can be decomposed into its partial fractions following the table above

Partial Fractions

TGM Original Questions

An algebraic fraction can be expressed in the form \( \frac{\mathrm{f}(x)}{\mathrm{g}(x)} \), where \(\mathrm{f}(x)\) and \(\mathrm{g}(x)\) are both polynomials.
If \( \mathrm{f}(x)\) has a degree \(<\) \(\mathrm{g}(x)\), it is a proper algebraic fraction
If \(\mathrm{f}(x)\) has a degree \( \geq\) \(\mathrm{g}(x)\), it is an improper algebraic fraction
Proper algebraic fractions can be expressed as partial fractions depending on the denominator. The expression of partial fractions are summarised in the table:

Denominator	Algebraic Fraction	Partial Fraction
Non-Repeating Linear Factors	\( \frac{px + q}{(ax + b)(cx + d)} \)	\( \frac{A}{ax + b} + \frac{B}{cx + d}\)
Repeating Linear Factors	\( \frac{px + q}{(ax + b)^2(cx + d)} \)	\(\frac{A}{ax + b} + \frac{B}{(ax + b)^2} + \frac{C}{cx + d} \)
Quadratic Factors that cannot be factorised	\( \frac{px + q}{(ax^2 + b)(cx + d)} \)	\( \frac{Ax + B}{ax^2 + b} + \frac{C}{cx + d} \)

The constants \(A, B, C,\) etc. can be found by equating the algebraic fraction to its partial fraction decomposition, clearing the denominators, and then comparing the coefficients of corresponding powers of \(x\).

When dealing with improper algebraic fractions, perform long division on the fraction \( \frac{\mathrm{f}(x)}{\mathrm{g}(x)} \) first, so that:
\( \frac{\mathrm{f}(x)}{\mathrm{g}(x)} = \mathrm{Q}(x) + \frac{\mathrm{R}(x)}{\mathrm{g}(x)}\)
The fraction \( \frac{\mathrm{R}(x)}{\mathrm{g}(x)}\) is now a proper algebraic fraction. It can be decomposed into its partial fractions following the table above

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