<ul class="bullet-list"><li data-list-item-id="item-1782211894971-fghf978hl"><div class="list-item-paragraph">An algebraic fraction can be expressed in the form \( \frac{\mathrm{f}(x)}{\mathrm{g}(x)} \), where&nbsp;\(\mathrm{f}(x)\) and \(\mathrm{g}(x)\) are both polynomials.<br>If&nbsp;\( \mathrm{f}(x)\) has a degree \(&lt;\) \(\mathrm{g}(x)\), it is a <strong>proper algebraic fraction</strong><br>If&nbsp;\(\mathrm{f}(x)\) has a degree \( \geq\) \(\mathrm{g}(x)\), it is an <strong>improper algebraic fraction</strong></div></li><li data-list-item-id="item-1782212381219-mq8lbun45"><div class="list-item-paragraph"><strong>Proper algebraic fractions</strong> can be expressed as partial fractions depending on the denominator. The expression of partial fractions are summarised in the table:</div></li></ul>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Denominator</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Algebraic Fraction</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Partial Fraction</strong></p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p><strong>Non-Repeating Linear Factors</strong></p></td><td style="text-align: center; --cell-justify: center"><p>\( \frac{px + q}{(ax + b)(cx + d)} \)</p></td><td style="text-align: center; --cell-justify: center"><p>\( \frac{A}{ax + b} + \frac{B}{cx + d}\)</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p><strong>Repeating Linear Factors</strong></p></td><td style="text-align: center; --cell-justify: center"><p>\( \frac{px + q}{(ax + b)^2(cx + d)} \)</p></td><td style="text-align: center; --cell-justify: center"><p>\(\frac{A}{ax + b} + \frac{B}{(ax + b)^2} + \frac{C}{cx + d} \)</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p><strong>Quadratic Factors that cannot be factorised</strong></p></td><td style="text-align: center; --cell-justify: center"><p>\( \frac{px + q}{(ax^2 + b)(cx + d)} \)</p></td><td style="text-align: center; --cell-justify: center"><p>\( \frac{Ax + B}{ax^2 + b} + \frac{C}{cx + d} \)</p></td></tr></tbody></table></div></div>
<ul class="bullet-list"><li data-list-item-id="item-1782215274570-tog2njwgq"><div class="list-item-paragraph">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 \(x\).</div></li></ul>
<ul class="bullet-list"><li data-list-item-id="item-1782212471776-k0flzm88m"><div class="list-item-paragraph">When dealing with <strong>improper algebraic fractions</strong>, perform long division on the fraction \( \frac{\mathrm{f}(x)}{\mathrm{g}(x)} \) first, so that:</div><div class="list-item-paragraph" style="text-align: center">\( \frac{\mathrm{f}(x)}{\mathrm{g}(x)} = \mathrm{Q}(x) + \frac{\mathrm{R}(x)}{\mathrm{g}(x)}\)</div></li><li data-list-item-id="item-1782213309225-v1vvgy5k6"><div class="list-item-paragraph">The fraction \( \frac{\mathrm{R}(x)}{\mathrm{g}(x)}\) is now a <strong>proper algebraic fraction</strong>. It can be decomposed into its partial fractions following the table above</div></li></ul>

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:

<p><strong>Algebraic Fraction</strong></p>

<p><strong>Non-Repeating Linear Factors</strong></p>

<p><strong>Repeating Linear Factors</strong></p>

<p><strong>Quadratic Factors that cannot be factorised</strong></p>

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

Partial Fractions