<ul class="bullet-list"><li data-list-item-id="item-1782208861472-9hza5f3l5"><div class="list-item-paragraph">Given a polynomial \( \mathrm{f}(x)\) is divided by a linear expression \(ax - b\), then its remainder is \( \mathrm{f} \left( \frac{b}{a} \right) \)<br>This follows from the division, where</div><div class="list-item-paragraph" style="text-align: center">\( \mathrm{f}(x) = (ax - b) \mathrm{Q}(x) + \mathrm{R}\)</div><div class="list-item-paragraph">where&nbsp;\( \mathrm{Q}(x)\) is the quotient and \(\mathrm{R}\) is the remainder</div></li></ul>
<p style="text-align: center">\( \mathrm{f} \ \left(\frac{b}{a} \right) = \mathrm{Q}\left(\frac{b}{a}\right) \times \left(a \times \frac{b}{a} - b \right) + \mathrm{R} \)<br>\( \mathrm{f} \left( \frac{b}{a} \right) = \mathrm{Q} \left( \frac{b}{a} \right) \times 0 + \mathrm{R} = \mathrm{R} \)</p>
<p><strong>Remainder when divided by higher degree polynomial</strong></p>
<ul class="bullet-list"><li data-list-item-id="item-1782209489391-xdr0qmnxn"><div class="list-item-paragraph">When&nbsp;\( \mathrm{f}(x) \) is divided by a polynomial of degree \(n\), its remainder will have a degree less than \(n\)</div></li><li data-list-item-id="item-1782208874639-j3xxu2dez"><div class="list-item-paragraph">For example when&nbsp;\( \mathrm{f}(x)\) is divided by a quadratic expression \(ax^2 + bx + c\), its remainder will be a linear expression \(dx + e\).</div></li><li data-list-item-id="item-1782214281407-g6wgnt0vx"><div class="list-item-paragraph">In summary:</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>Divisior</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Remainder</strong></p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>\(ax + b\) degree \(1\)</p></td><td style="text-align: center; --cell-justify: center"><p>Constant</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>\(ax^2 + bx + c\) degree \(2\)</p></td><td style="text-align: center; --cell-justify: center"><p>Linear expression \(dx + e\) or constant</p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>\(ax^3 + bx^2 + cx + d\) degree \(3\)</p></td><td style="text-align: center; --cell-justify: center"><p>Quadratic expression \(ex^2 + fx + g\) or lower degree</p></td></tr></tbody></table></div></div>

Given a polynomial \( \mathrm{f}(x)\) is divided by a linear expression \(ax - b\), then its remainder is \( \mathrm{f} \left( \frac{b}{a} \right) \)
This follows from the division, where

\( \mathrm{f}(x) = (ax - b) \mathrm{Q}(x) + \mathrm{R}\)

where \( \mathrm{Q}(x)\) is the quotient and \(\mathrm{R}\) is the remainder

\( \mathrm{f} \ \left(\frac{b}{a} \right) = \mathrm{Q}\left(\frac{b}{a}\right) \times \left(a \times \frac{b}{a} - b \right) + \mathrm{R} \)
\( \mathrm{f} \left( \frac{b}{a} \right) = \mathrm{Q} \left( \frac{b}{a} \right) \times 0 + \mathrm{R} = \mathrm{R} \)

**Remainder when divided by higher degree polynomial**

When \( \mathrm{f}(x) \) is divided by a polynomial of degree \(n\), its remainder will have a degree less than \(n\)

For example when \( \mathrm{f}(x)\) is divided by a quadratic expression \(ax^2 + bx + c\), its remainder will be a linear expression \(dx + e\).

<p>Linear expression \(dx + e\) or constant</p>

<p>\(ax^3 + bx^2 + cx + d\) degree \(3\)</p>

<p>Quadratic expression \(ex^2 + fx + g\) or lower degree</p>

Remainder Theorem

Remainder Theorem