All denominators are nonzero. For the linear-factor rows, $a,c\ne0$ and $ad-bc\ne0$. For $(ax^2+b)(cx+d)$, assume $c\ne0$ and $ab>0$, so the quadratic is irreducible over the reals. The degree comparison for a proper fraction concerns a nonzero numerator; a zero numerator gives the zero fraction. 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 $\deg f<\deg g$, it is a proper algebraic fraction If $\deg f\geq\deg g$, 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)}$ If $R\ne0$, then $\deg R<\deg g$, so $R/g$ is proper and can be decomposed using the table. If $R=0$, the fraction equals the polynomial quotient.

Partial Fractions

TGM Original Questions

All denominators are nonzero. For the linear-factor rows, $a,c\ne0$ and $ad-bc\ne0$. For $(ax^2+b)(cx+d)$, assume $c\ne0$ and $ab>0$, so the quadratic is irreducible over the reals. The degree comparison for a proper fraction concerns a nonzero numerator; a zero numerator gives the zero fraction.

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 $\deg f<\deg g$, it is a proper algebraic fraction
If $\deg f\geq\deg g$, 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)}$
If $R\ne0$, then $\deg R<\deg g$, so $R/g$ is proper and can be decomposed using the table. If $R=0$, the fraction equals the polynomial quotient.

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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	\( \frac{px + q}{(ax^2 + b)(cx + d)} \)	\( \frac{Ax + B}{ax^2 + b} + \frac{C}{cx + d} \)