Polynomial division & factor theorem

<p>For a polynomial \(P\) and linear divisor \(ax+b\), \(a\ne0\):</p>
<div class="content-box-placement content-box-placement--full content-box-placement--align-center" style="display:flex;justify-content:stretch;max-width:100%"><div class="content-box content-box--emphasis" data-content-box-preset="emphasis" style="box-sizing:border-box;border-color:var(--content-box-border, #1E6FBA);border-style:solid;border-width:0px;border-radius:0px;background-color:var(--content-box-background, #F5F8FC);color:var(--content-box-foreground, #111827);padding:12px 16px;font-family:'EB Garamond', serif;font-size:16px;line-height:1.35;width:100%;max-width:100%;border-left-width:3px"><p style="text-align: center">\(\text{Remainder}=P\!\left(-\dfrac ba\right)\)</p></div></div>
<p>Set the divisor equal to zero to find the value to substitute.</p>
<p class="paragraph-header">Examples</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 22.2222%;"><col style="width: 25.9259%;"><col style="width: 51.8519%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Divisor</strong></th><th style="text-align: center; --cell-justify: center"><strong>Remainder</strong></th><th style="text-align: center; --cell-justify: center"><strong>Example: \(\mathrm{f}(x)=2x^3-x+3\)</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(x-c\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center">\(ax-b\)</td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td><td class="table-cell--fill-in" style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></div>

For a polynomial \(P\) and linear divisor \(ax+b\), \(a\ne0\):

\(\text{Remainder}=P\!\left(-\dfrac ba\right)\)

Set the divisor equal to zero to find the value to substitute.

Divided by \(x-2\):
\(\begin{aligned} \mathrm{f}(2)&=16-2+3 \\ &=17 \end{aligned}\)

\(\mathrm{f}\!\left(\dfrac{b}{a}\right)\)

Divided by \(2x+1\):
\(\begin{aligned} \mathrm{f}\!\left(-\dfrac12\right)&=-\dfrac14+\dfrac12+3 \\ &=3\dfrac14 \end{aligned}\)

Divisor	Remainder	Example: \(\mathrm{f}(x)=2x^3-x+3\)
\(x-c\)
\(ax-b\)

Remainder Theorem

Remainder Theorem

Video Solution:

Solution:

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