<ul class="bullet-list"><li data-list-item-id="item-1782209743142-mxmkosfci"><div class="list-item-paragraph">Given a polynomial \( \mathrm{f}(x)\) is divided by a linear factor \(ax - b\), then&nbsp;\( \mathrm{f} \left( \frac{b}{a} \right) = 0 \)<br>This follows from the division algorithm, where</div><div class="list-item-paragraph" style="text-align: center">\( \mathrm{f}(x) = (ax - b) \mathrm{Q}(x)\)</div><div class="list-item-paragraph">where&nbsp;\( \mathrm{Q}(x)\) is the quotient and \(\mathrm{R}\) is the remainder<br>*Note there is no remainder from the division since&nbsp;\( \mathrm{f}(x)\) is divided by its factor<br>Then,</div><div class="list-item-paragraph" style="text-align: center">\( \mathrm{f} \ \left(\frac{b}{a} \right) = \mathrm{Q}\left(\frac{b}{a}\right) \times \left(a \times \frac{b}{a} - b \right) \)<br>\( \mathrm{f} \left( \frac{b}{a} \right) = \mathrm{Q} \left( \frac{b}{a} \right) \times 0 = 0 \)</div></li></ul>

Given a polynomial \( \mathrm{f}(x)\) is divided by a linear factor \(ax - b\), then \( \mathrm{f} \left( \frac{b}{a} \right) = 0 \)
This follows from the division algorithm, where

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

where \( \mathrm{Q}(x)\) is the quotient and \(\mathrm{R}\) is the remainder
*Note there is no remainder from the division since \( \mathrm{f}(x)\) is divided by its factor
Then,

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

Factor Theorem

Factor Theorem