Polynomial division & factor theorem

<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">\(\mathrm{P}(x)=\mathrm{D}(x)\mathrm{Q}(x)+\mathrm{R}(x)\)</p></div></div>
<p>\(\mathrm{D}\) is a nonzero polynomial. The remainder is zero or has degree less than \(\mathrm{D}\).</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">\(\dfrac{\mathrm{P}(x)}{\mathrm{D}(x)}=\mathrm{Q}(x)+\dfrac{\mathrm{R}(x)}{\mathrm{D}(x)}\qquad \mathrm{D}(x)\ne0\)</p></div></div>
<p>Arrange terms in descending powers, inserting zero coefficients for missing powers. Divide the leading terms, multiply, subtract and repeat.</p>
<p class="paragraph-header">Example: divide \(3x^3-4x^2+5x+4\) by \(x^2-3x\)</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 540px;"><colgroup><col style="width: 20.3704%;"><col style="width: 36.1111%;"><col style="width: 43.5185%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><strong>Quotient term</strong></th><th style="text-align: center; --cell-justify: center"><strong>Subtract</strong></th><th style="text-align: center; --cell-justify: center"><strong>Remainder</strong></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">\(3x\)</td><td style="text-align: center; --cell-justify: center">\(3x(x^2-3x)\)</td><td style="text-align: center; --cell-justify: center">\(5x^2+5x+4\)</td></tr><tr><td style="text-align: center; --cell-justify: center">\(5\)</td><td style="text-align: center; --cell-justify: center">\(5(x^2-3x)\)</td><td style="text-align: center; --cell-justify: center">\(20x+4\)</td></tr></tbody></table></div></div>
<p>Quotient: \(3x+5\). Remainder: \(20x+4\).</p>
<p>Check: \(3x^3-4x^2+5x+4=(x^2-3x)(3x+5)+(20x+4)\).</p>
<p>The remainder theorem finds only the remainder; use division when the quotient is also needed.</p>

\(\mathrm{P}(x)=\mathrm{D}(x)\mathrm{Q}(x)+\mathrm{R}(x)\)

\(\mathrm{D}\) is a nonzero polynomial. The remainder is zero or has degree less than \(\mathrm{D}\).

\(\dfrac{\mathrm{P}(x)}{\mathrm{D}(x)}=\mathrm{Q}(x)+\dfrac{\mathrm{R}(x)}{\mathrm{D}(x)}\qquad \mathrm{D}(x)\ne0\)

Arrange terms in descending powers, inserting zero coefficients for missing powers. Divide the leading terms, multiply, subtract and repeat.

Example: divide \(3x^3-4x^2+5x+4\) by \(x^2-3x\)

Quotient: \(3x+5\). Remainder: \(20x+4\).

Check: \(3x^3-4x^2+5x+4=(x^2-3x)(3x+5)+(20x+4)\).

The remainder theorem finds only the remainder; use division when the quotient is also needed.

Long Division of Polynomials

Long Division of Polynomials

Quotient term	Subtract	Remainder
\(3x\)	\(3x(x^2-3x)\)	\(5x^2+5x+4\)
\(5\)	\(5(x^2-3x)\)	\(20x+4\)