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<head>
<title>DLMF: 1.11 Zeros of Polynomials</title>
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<text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default">
<div class="ltx_page_navbar">
<div class="ltx_page_navlogo"></dd>
</dl>
</div>
</div>
<div id="Px1.p1" class="ltx_para">
<p class="ltx_p">Let</p>
<table id="E1" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.1</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E1.png" altimg-height="28px" altimg-valign="-7px" altimg-width="317px" alttext="f(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\dots+a_{0}." display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow><mo>+</mo><mrow><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E1.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then</p>
<table id="E2" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.2</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="28px" altimg-valign="-7px" altimg-width="452px" alttext="f(z)=(z-\alpha)(b_{n}z^{n-1}+b_{n-1}z^{n-2}+\dots+b_{1})+b_{0}," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><mi>α</mi></mrow><mo stretchy="false">)</mo></mrow><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mrow><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>+</mo><msub><mi>b</mi><mn>0</mn></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E2.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m90.png" altimg-height="21px" altimg-valign="-5px" altimg-width="72px" alttext="b_{n}=a_{n}" display="inline"><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math>,
</p>
<table id="E3" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.3</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="24px" altimg-valign="-7px" altimg-width="153px" alttext="b_{k}=\alpha b_{k+1}+a_{k}," display="block"><mrow><mrow><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo>=</mo><mrow><mrow><mi>α</mi><mo></mo><msub><mi>b</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m117.png" altimg-height="21px" altimg-valign="-6px" altimg-width="198px" alttext="k=n-1,n-2,\dots,0" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mn>0</mn></mrow></mrow></math>,</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E3.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<table id="E4" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.4</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="f(\alpha)=b_{0}." display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msub><mi>b</mi><mn>0</mn></msub></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px2.p1" class="ltx_para">
<p class="ltx_p">With <math class="ltx_Math" altimg="m89.png" altimg-height="21px" altimg-valign="-5px" altimg-width="23px" alttext="b_{k}" display="inline"><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></math> as in () let
<math class="ltx_Math" altimg="m92.png" altimg-height="16px" altimg-valign="-5px" altimg-width="72px" alttext="c_{n}=a_{n}" display="inline"><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>=</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> and
</p>
<table id="E5" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.5</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E5.png" altimg-height="24px" altimg-valign="-7px" altimg-width="151px" alttext="c_{k}=\alpha c_{k+1}+b_{k}," display="block"><mrow><mrow><msub><mi>c</mi><mi href="./1.1#p2.t1.r4">k</mi></msub><mo>=</mo><mrow><mrow><mi>α</mi><mo></mo><msub><mi>c</mi><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><mo>+</mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr class="ltx_eqn_row"><td class="ltx_eqn_cell ltx_align_right" colspan="4"><span class="ltx_constraint"><math class="ltx_Math" altimg="m118.png" altimg-height="21px" altimg-valign="-6px" altimg-width="198px" alttext="k=n-1,n-2,\dots,1" display="inline"><mrow><mi href="./1.1#p2.t1.r4">k</mi><mo>=</mo><mrow><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow><mo>,</mo><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><mn>1</mn></mrow></mrow></math>.</span></td></tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E5.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Then
</p>
<table id="E6" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.6</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="26px" altimg-valign="-7px" altimg-width="102px" alttext="f^{\prime}(\alpha)=c_{1}." display="block"><mrow><mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo></mo><mrow><mo stretchy="false">(</mo><mi>α</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><msub><mi>c</mi><mn>1</mn></msub></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px2.p2" class="ltx_para">
<p class="ltx_p">More generally, for polynomials <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> and <math class="ltx_Math" altimg="m109.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="g(z)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, there are polynomials <math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="q(z)" display="inline"><mrow><mi href="./1.11#Px2.p2">q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>
and <math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="r(z)" display="inline"><mrow><mi href="./1.11#Px2.p2">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>, found by equating coefficients, such that</p>
<table id="E7" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.7</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="206px" alttext="f(z)=g(z)q(z)+r(z)," display="block"><mrow><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo></mo><mrow><mi href="./1.11#Px2.p2">q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>+</mo><mrow><mi href="./1.11#Px2.p2">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E7.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.11#Px2.p2" title="Extended Horner Scheme ‣ §1.11(i) Division Algorithm ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m123.png" altimg-height="23px" altimg-valign="-7px" altimg-width="40px" alttext="q(z)" display="inline"><mrow><mi href="./1.11#Px2.p2">q</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: polynomial</a> and
<a href="./1.11#Px2.p2" title="Extended Horner Scheme ‣ §1.11(i) Division Algorithm ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m126.png" altimg-height="23px" altimg-valign="-7px" altimg-width="39px" alttext="r(z)" display="inline"><mrow><mi href="./1.11#Px2.p2">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>: polynomial</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m46.png" altimg-height="23px" altimg-valign="-7px" altimg-width="205px" alttext="0\leq\deg r(z)<\deg g(z)" display="inline"><mrow><mn>0</mn><mo>≤</mo><mrow><mi>deg</mi><mo></mo><mrow><mi href="./1.11#Px2.p2">r</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow><mo><</mo><mrow><mi>deg</mi><mo></mo><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow></math>.</p>
</div>
</section>
</section>
<section id="ii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.11(ii) </span>Elementary Properties</h2>
<div id="SS2.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="SS2.p1" class="ltx_para">
<p class="ltx_p">A polynomial of degree <math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math> with real or complex coefficients has exactly <math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>
real or complex zeros counting multiplicity. Every <em class="ltx_emph ltx_font_italic">monic</em> (coefficient of
highest power is one) polynomial of odd degree with real coefficients has at
least one real zero with sign opposite to that of the constant term. A monic
polynomial of even degree with real coefficients has at least two zeros of
opposite signs when the constant term is negative.</p>
</div>
<section id="Px3" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Descartes’ Rule of Signs</h3>
<div id="Px3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px3.p1" class="ltx_para">
<p class="ltx_p">The number of positive zeros of a polynomial with real coefficients cannot
exceed the number of times the coefficients change sign, and the two numbers
have same parity. A similar relation holds for the changes in sign of the
coefficients of <math class="ltx_Math" altimg="m96.png" altimg-height="23px" altimg-valign="-7px" altimg-width="57px" alttext="f(-z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math>, and hence for the number of negative zeros of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
</section>
<section id="Px4" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Example</h3>
<div id="Px4.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px4.p1" class="ltx_para">
<table id="E8" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.8</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="25px" altimg-valign="-7px" altimg-width="44px" alttext="\displaystyle f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="27px" altimg-valign="-6px" altimg-width="184px" alttext="\displaystyle=z^{8}+10z^{3}+z-4," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>8</mn></msup><mo>+</mo><mrow><mn>10</mn><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup></mrow><mo>+</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>-</mo><mn>4</mn></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="25px" altimg-valign="-7px" altimg-width="59px" alttext="\displaystyle f(-z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="25px" altimg-valign="-4px" altimg-width="184px" alttext="\displaystyle=z^{8}-10z^{3}-z-4." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>8</mn></msup><mo>-</mo><mrow><mn>10</mn><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup></mrow><mo>-</mo><mi href="./1.1#p2.t1.r2">z</mi><mo>-</mo><mn>4</mn></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E8.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Both polynomials have one change of sign; hence for each polynomial there is
one positive zero, one negative zero, and six complex zeros.</p>
</div>
<div id="Px4.p2" class="ltx_para">
<p class="ltx_p">Next, let <math class="ltx_Math" altimg="m98.png" altimg-height="25px" altimg-valign="-7px" altimg-width="310px" alttext="f(z)=a_{n}z^{n}+a_{n-1}z^{n-1}+\dots+a_{0}" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow><mo>+</mo><mrow><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi>a</mi><mn>0</mn></msub></mrow></mrow></math>. The zeros of
<math class="ltx_Math" altimg="m134.png" altimg-height="25px" altimg-valign="-7px" altimg-width="328px" alttext="z^{n}f(1/z)=a_{0}z^{n}+a_{1}z^{n-1}+\dots+a_{n}" display="inline"><mrow><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup><mo></mo><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mn>1</mn><mo>/</mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow><mo>=</mo><mrow><mrow><msub><mi>a</mi><mn>0</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msup></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow></math> are reciprocals of the zeros
of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>.</p>
</div>
<div id="Px4.p3" class="ltx_para">
<p class="ltx_p">The <em class="ltx_emph ltx_font_italic">discriminant</em> of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> is defined by
</p>
<table id="E9" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.9</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E9.png" altimg-height="55px" altimg-valign="-31px" altimg-width="222px" alttext="D=a_{n}^{2n-2}\prod_{j<k}(z_{j}-z_{k})^{2}," display="block"><mrow><mrow><mi href="./1.11#E9">D</mi><mo>=</mo><mrow><msubsup><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi><mrow><mrow><mn>2</mn><mo></mo><mi href="./1.1#p2.t1.r5">n</mi></mrow><mo>-</mo><mn>2</mn></mrow></msubsup><mo></mo><mrow><munder><mo largeop="true" movablelimits="false" symmetric="true">∏</mo><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo><</mo><mi href="./1.1#p2.t1.r4">k</mi></mrow></munder><msup><mrow><mo stretchy="false">(</mo><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo>-</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E9.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.11#E9">D</mi></math>: discriminant of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math> (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">where <math class="ltx_Math" altimg="m135.png" altimg-height="16px" altimg-valign="-6px" altimg-width="114px" alttext="z_{1},z_{2},\dots,z_{n}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>,</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>,</mo><mi mathvariant="normal">…</mi><mo>,</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math> are the zeros of <math class="ltx_Math" altimg="m103.png" altimg-height="23px" altimg-valign="-7px" altimg-width="42px" alttext="f(z)" display="inline"><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow></math>. The <em class="ltx_emph ltx_font_italic">elementary
symmetric functions</em>
of the zeros are (with <math class="ltx_Math" altimg="m87.png" altimg-height="21px" altimg-valign="-6px" altimg-width="62px" alttext="a_{n}\not=0" display="inline"><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub><mo>≠</mo><mn>0</mn></mrow></math>)
</p>
</div>
<div id="Px4.p4" class="ltx_para">
<table id="E10" class="ltx_equationgroup ltx_eqn_table">
<tr id="E10X" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="4" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.10</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m41.png" altimg-height="20px" altimg-valign="-5px" altimg-width="159px" alttext="\displaystyle z_{1}+z_{2}+\dots+z_{n}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo>+</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo>+</mo><mi mathvariant="normal">…</mi><mo>+</mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m13.png" altimg-height="25px" altimg-valign="-7px" altimg-width="121px" alttext="\displaystyle=-a_{n-1}/a_{n}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="E10Xa" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="55px" altimg-valign="-31px" altimg-width="122px" alttext="\displaystyle\sum_{1\leq j<k\leq n}z_{j}z_{k}" display="inline"><mrow><mstyle displaystyle="true"><munder><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mn>1</mn><mo>≤</mo><mi href="./1.1#p2.t1.r4">j</mi><mo><</mo><mi href="./1.1#p2.t1.r4">k</mi><mo>≤</mo><mi href="./1.1#p2.t1.r5">n</mi></mrow></munder></mstyle><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r4">k</mi></msub></mrow></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m20.png" altimg-height="25px" altimg-valign="-7px" altimg-width="106px" alttext="\displaystyle=a_{n-2}/a_{n}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msub><mi>a</mi><mrow><mi href="./1.1#p2.t1.r5">n</mi><mo>-</mo><mn>2</mn></mrow></msub><mo>/</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="E10Xb" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="E10c.png" altimg-height="36px" altimg-valign="-2px" altimg-width="12px" alttext="\displaystyle\mathrel{\vdots}" display="inline"><mo>⋮</mo></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="E10Xc" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m42.png" altimg-height="17px" altimg-valign="-5px" altimg-width="92px" alttext="\displaystyle z_{1}z_{2}\cdots z_{n}" display="inline"><mrow><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>1</mn></msub><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msub><mo></mo><mi mathvariant="normal">⋯</mi><mo></mo><msub><mi href="./1.1#p2.t1.r2">z</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="135px" alttext="\displaystyle=(-1)^{n}a_{0}/a_{n}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mrow><mo>-</mo><mn>1</mn></mrow><mo stretchy="false">)</mo></mrow><mi href="./1.1#p2.t1.r5">n</mi></msup><mo></mo><msub><mi>a</mi><mn>0</mn></msub></mrow><mo>/</mo><msub><mi>a</mi><mi href="./1.1#p2.t1.r5">n</mi></msub></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E10.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd>
<a href="./1.1#p2.t1.r2" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m131.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="z" display="inline"><mi href="./1.1#p2.t1.r2">z</mi></math>: variable</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m113.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a>,
<a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m119.png" altimg-height="18px" altimg-valign="-2px" altimg-width="15px" alttext="k" display="inline"><mi href="./1.1#p2.t1.r4">k</mi></math>: integer</a> and
<a href="./1.1#p2.t1.r5" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m120.png" altimg-height="13px" altimg-valign="-2px" altimg-width="16px" alttext="n" display="inline"><mi href="./1.1#p2.t1.r5">n</mi></math>: nonnegative integer</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
</section>
</section>
<section id="iii" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">§1.11(iii) </span>Polynomials of Degrees Two, Three, and Four</h2>
<div id="SS3.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px5.p1" class="ltx_para">
<p class="ltx_p">The roots of <math class="ltx_Math" altimg="m88.png" altimg-height="21px" altimg-valign="-4px" altimg-width="146px" alttext="az^{2}+bz+c=0" display="inline"><mrow><mrow><mrow><mi>a</mi><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mi>b</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>c</mi></mrow><mo>=</mo><mn>0</mn></mrow></math> are</p>
<table id="E11" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.11</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center" colspan="2"><math class="ltx_Math" altimg="E11a.png" altimg-height="31px" altimg-valign="-9px" altimg-width="73px" alttext="\frac{-b\pm\sqrt{D}}{2a}," display="inline"><mrow><mstyle displaystyle="true"><mfrac><mrow><mrow><mo>-</mo><mi>b</mi></mrow><mo>±</mo><msqrt><mi href="./1.11#E12">D</mi></msqrt></mrow><mrow><mn>2</mn><mo></mo><mi>a</mi></mrow></mfrac></mstyle><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="19px" altimg-valign="-2px" altimg-width="23px" alttext="\displaystyle D" display="inline"><mi href="./1.11#E12">D</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m21.png" altimg-height="25px" altimg-valign="-4px" altimg-width="104px" alttext="\displaystyle=b^{2}-4ac." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mrow><mn>4</mn><mo></mo><mi>a</mi><mo></mo><mi>c</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E11.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Symbols:</dt>
<dd><a href="./1.11#E12" title="(1.11.12) ‣ Cubic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.11#E12">D</mi></math>: discriminant</a></dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">The sum and product of the roots are respectively <math class="ltx_Math" altimg="m45.png" altimg-height="23px" altimg-valign="-7px" altimg-width="49px" alttext="-b/a" display="inline"><mrow><mo>-</mo><mrow><mi>b</mi><mo>/</mo><mi>a</mi></mrow></mrow></math> and <math class="ltx_Math" altimg="m91.png" altimg-height="23px" altimg-valign="-7px" altimg-width="33px" alttext="c/a" display="inline"><mrow><mi>c</mi><mo>/</mo><mi>a</mi></mrow></math>.</p>
</div>
</section>
<section id="Px6" class="ltx_paragraph">
<h3 class="ltx_title ltx_title_paragraph">Cubic Equations</h3>
<div id="Px6.info" class="ltx_metadata ltx_info">
</dd>
</dl>
</div>
</div>
<div id="Px6.p1" class="ltx_para">
<p class="ltx_p">Set <math class="ltx_Math" altimg="m129.png" altimg-height="27px" altimg-valign="-9px" altimg-width="103px" alttext="z=w-\tfrac{1}{3}a" display="inline"><mrow><mi href="./1.1#p2.t1.r2">z</mi><mo>=</mo><mrow><mi href="./1.1#p2.t1.r3">w</mi><mo>-</mo><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mi>a</mi></mrow></mrow></mrow></math> to reduce <math class="ltx_Math" altimg="m99.png" altimg-height="25px" altimg-valign="-7px" altimg-width="217px" alttext="f(z)=z^{3}+az^{2}+bz+c" display="inline"><mrow><mrow><mi>f</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r2">z</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>3</mn></msup><mo>+</mo><mrow><mi>a</mi><mo></mo><msup><mi href="./1.1#p2.t1.r2">z</mi><mn>2</mn></msup></mrow><mo>+</mo><mrow><mi>b</mi><mo></mo><mi href="./1.1#p2.t1.r2">z</mi></mrow><mo>+</mo><mi>c</mi></mrow></mrow></math> to
<math class="ltx_Math" altimg="m105.png" altimg-height="25px" altimg-valign="-7px" altimg-width="178px" alttext="g(w)=w^{3}+pw+q" display="inline"><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mrow><msup><mi href="./1.1#p2.t1.r3">w</mi><mn>3</mn></msup><mo>+</mo><mrow><mi href="./1.11#Px8.p1">p</mi><mo></mo><mi href="./1.1#p2.t1.r3">w</mi></mrow><mo>+</mo><mi href="./1.11#Px8.p1">q</mi></mrow></mrow></math>, with <math class="ltx_Math" altimg="m121.png" altimg-height="25px" altimg-valign="-7px" altimg-width="139px" alttext="p=(3b-a^{2})/3" display="inline"><mrow><mi href="./1.11#Px8.p1">p</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mn>3</mn><mo></mo><mi>b</mi></mrow><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>3</mn></mrow></mrow></math>, <math class="ltx_Math" altimg="m124.png" altimg-height="25px" altimg-valign="-7px" altimg-width="222px" alttext="q=(2a^{3}-9ab+27c)/27" display="inline"><mrow><mi href="./1.11#Px8.p1">q</mi><mo>=</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><mrow><mn>2</mn><mo></mo><msup><mi>a</mi><mn>3</mn></msup></mrow><mo>-</mo><mrow><mn>9</mn><mo></mo><mi>a</mi><mo></mo><mi>b</mi></mrow></mrow><mo>+</mo><mrow><mn>27</mn><mo></mo><mi>c</mi></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>/</mo><mn>27</mn></mrow></mrow></math>. The
discriminant of <math class="ltx_Math" altimg="m108.png" altimg-height="23px" altimg-valign="-7px" altimg-width="45px" alttext="g(w)" display="inline"><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow></math> is</p>
<table id="E12" class="ltx_equation ltx_eqn_table">
<tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.11.12</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="27px" altimg-valign="-6px" altimg-width="162px" alttext="D=-4p^{3}-27q^{2}." display="block"><mrow><mrow><mi href="./1.11#E12">D</mi><mo>=</mo><mrow><mrow><mo>-</mo><mrow><mn>4</mn><mo></mo><msup><mi href="./1.11#Px8.p1">p</mi><mn>3</mn></msup></mrow></mrow><mo>-</mo><mrow><mn>27</mn><mo></mo><msup><mi href="./1.11#Px8.p1">q</mi><mn>2</mn></msup></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="4">
<div id="E12.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd><span class="ltx_text"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.11#E12">D</mi></math>: discriminant (locally)</span></dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m122.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./1.11#Px8.p1">p</mi></math></a> and
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./1.11#Px8.p1">q</mi></math></a>
</dd>
<dt>A&S Ref:</dt>
<dd> <span class="ltx_origref"><span class="ltx_tag">3.8.1</span></span>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
<p class="ltx_p">Let</p>
<table id="E13" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.13</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="19px" altimg-valign="-2px" altimg-width="21px" alttext="\displaystyle A" display="inline"><mi href="./1.11#E13">A</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="42px" altimg-valign="-12px" altimg-width="196px" alttext="\displaystyle=\sqrt[3]{-\tfrac{27}{2}q+\tfrac{3}{2}\sqrt{-3D}}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mroot><mrow><mrow><mo>-</mo><mrow><mfrac><mn>27</mn><mn>2</mn></mfrac><mo></mo><mi href="./1.11#Px8.p1">q</mi></mrow></mrow><mo>+</mo><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo></mo><msqrt><mrow><mo>-</mo><mrow><mn>3</mn><mo></mo><mi href="./1.11#E12">D</mi></mrow></mrow></msqrt></mrow></mrow><mn>3</mn></mroot></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="19px" altimg-valign="-2px" altimg-width="22px" alttext="\displaystyle B" display="inline"><mi href="./1.11#E13">B</mi></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="25px" altimg-valign="-7px" altimg-width="92px" alttext="\displaystyle=-3p/A." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mrow><mrow><mn>3</mn><mo></mo><mi href="./1.11#Px8.p1">p</mi></mrow><mo>/</mo><mi href="./1.11#E13">A</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr><td class="ltx_align_right" colspan="5">
<div id="E13.info" class="ltx_metadata ltx_info">
<div class="ltx_infocontent">
<dl>
<dt>Defines:</dt>
<dd>
<span class="ltx_text"><math class="ltx_Math" altimg="m56.png" altimg-height="18px" altimg-valign="-2px" altimg-width="19px" alttext="A" display="inline"><mi href="./1.11#E13">A</mi></math> (locally)</span> and
<span class="ltx_text"><math class="ltx_Math" altimg="m58.png" altimg-height="18px" altimg-valign="-2px" altimg-width="20px" alttext="B" display="inline"><mi href="./1.11#E13">B</mi></math> (locally)</span>
</dd>
<dt>Symbols:</dt>
<dd>
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m122.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="p" display="inline"><mi href="./1.11#Px8.p1">p</mi></math></a>,
<a href="./1.11#Px8.p1" title="Quartic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m125.png" altimg-height="16px" altimg-valign="-6px" altimg-width="14px" alttext="q" display="inline"><mi href="./1.11#Px8.p1">q</mi></math></a> and
<a href="./1.11#E12" title="(1.11.12) ‣ Cubic Equations ‣ §1.11(iii) Polynomials of Degrees Two, Three, and Four ‣ §1.11 Zeros of Polynomials ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m59.png" altimg-height="18px" altimg-valign="-2px" altimg-width="21px" alttext="D" display="inline"><mi href="./1.11#E12">D</mi></math>: discriminant</a>
</dd>
</dl>
</div>
</div>
</td></tr>
</table>
</div>
<div id="Px6.p2" class="ltx_para">
<p class="ltx_p">The roots of <math class="ltx_Math" altimg="m104.png" altimg-height="23px" altimg-valign="-7px" altimg-width="81px" alttext="g(w)=0" display="inline"><mrow><mrow><mi>g</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi href="./1.1#p2.t1.r3">w</mi><mo stretchy="false">)</mo></mrow></mrow><mo>=</mo><mn>0</mn></mrow></math> are</p>
<table id="E14" class="ltx_equationgroup ltx_eqn_table">
<tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.11.14</span></td>
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14a.png" altimg-height="27px" altimg-valign="-9px" altimg-width="93px" alttext="\tfrac{1}{3}(A+B)," display="inline"><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./1.11#E13">A</mi><mo>+</mo><mi href="./1.11#E13">B</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14b.png" altimg-height="27px" altimg-valign="-9px" altimg-width="123px" alttext="\tfrac{1}{3}(\rho A+\rho^{2}B)," display="inline"><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><mo></mo><mrow><mo stretchy="false">(</mo><mrow><mrow><mi href="./1.11#Px6.p2">ρ</mi><mo></mo><mi href="./1.11#E13">A</mi></mrow><mo>+</mo><mrow><msup><mi href="./1.11#Px6.p2">ρ</mi><mn>2</mn></msup><mo></mo><mi href="./1.11#E13">B</mi></mrow></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr>
<tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14c.png" altimg-height="27px" altimg-valign="-9px" altimg-width="123px" alttext="\tfrac{1}{3}(\rho^{2}A+\rho B)," display="inline"><mrow><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac