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004 Sample Final A, Problem 8 - Revision history
2026-04-22T17:28:28Z
Revision history for this page on the wiki
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https://wiki.math.ucr.edu/index.php?title=004_Sample_Final_A,_Problem_8&diff=836&oldid=prev
MathAdmin at 17:18, 2 June 2015
2015-06-02T17:18:30Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:18, 2 June 2015</td>
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<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:: <span class="exam"> a) List all the possible rational zeros of the function <math>f(x)=x^4-4x^3-7x^2+34x-24.</math> <br></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:: <span class="exam"> a) List all the possible rational zeros of the function <math>f(x)=x^4-4x^3-7x^2+34x-24.</math<ins class="diffchange diffchange-inline">> </span</ins>><br></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>:: b) Find all the zeros, that is, solve <math>f(x) = 0</math></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>:: <ins class="diffchange diffchange-inline"><span class="exam"> </ins>b) Find all the zeros, that is, solve <math>f(x) = 0</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| class="mw-collapsible mw-collapsed" style = "text-align:left;"</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| class="mw-collapsible mw-collapsed" style = "text-align:left;"</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>! Foundations</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>! Foundations</div></td></tr>
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MathAdmin
https://wiki.math.ucr.edu/index.php?title=004_Sample_Final_A,_Problem_8&diff=835&oldid=prev
MathAdmin at 17:18, 2 June 2015
2015-06-02T17:18:10Z
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<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:18, 2 June 2015</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><span class="exam"> a) List all the possible rational zeros of the function <math>f(x)=x^4-4x^3-7x^2+34x-24.</math> <br></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:: </ins><span class="exam"> a) List all the possible rational zeros of the function <math>f(x)=x^4-4x^3-7x^2+34x-24.</math> <br></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>b) Find all the zeros, that is, solve <math>f(x) = 0</math></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">:: </ins>b) Find all the zeros, that is, solve <math>f(x) = 0</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| class="mw-collapsible mw-collapsed" style = "text-align:left;"</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| class="mw-collapsible mw-collapsed" style = "text-align:left;"</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>! Foundations</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>! Foundations</div></td></tr>
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MathAdmin
https://wiki.math.ucr.edu/index.php?title=004_Sample_Final_A,_Problem_8&diff=814&oldid=prev
MathAdmin: Created page with "<span class="exam"> a) List all the possible rational zeros of the function <math>f(x)=x^4-4x^3-7x^2+34x-24.</math> <br> b) Find all the zeros, that is, solve <math>f(x) = 0</..."
2015-06-01T05:48:14Z
<p>Created page with "<span class="exam"> a) List all the possible rational zeros of the function <math>f(x)=x^4-4x^3-7x^2+34x-24.</math> <br> b) Find all the zeros, that is, solve <math>f(x) = 0</..."</p>
<p><b>New page</b></p><div><span class="exam"> a) List all the possible rational zeros of the function <math>f(x)=x^4-4x^3-7x^2+34x-24.</math> <br><br />
b) Find all the zeros, that is, solve <math>f(x) = 0</math><br />
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"<br />
! Foundations<br />
|-<br />
|If <math>f(x)=x^4+bx^3+cx^2+dx+e</math>, what does the rational roots tell us are the possible roots of <math>f(x)</math>? <br />
|-<br />
|Answer:<br />
|-<br />
|The rational roots tells us that the possible roots of <math>f(x)</math> are <math>\pm k</math> where <math>k</math> is a divisor of <math>e</math>.<br />
|}<br />
<br />
<br />
Solution:<br />
<br />
{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"<br />
! Step 1:<br />
|-<br />
|By the rational roots test, the possible roots of <math>f(x)</math> are <math>\pm\{1,2,3,4,6,8,12,24\}</math>.<br />
|}<br />
<br />
{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"<br />
! Step 2:<br />
|-<br />
|Using synthetic division, we test 1 as a root of <math>f(x)</math>. We get a remainder of 0. So, we have that 1 is a root of <math>f(x)</math>. <br />
|-<br />
|By synthetic division, <math>f(x)=(x-1)(x^3-3x^2-10x+24)</math>.<br />
|}<br />
<br />
{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"<br />
! Step 3:<br />
|-<br />
|Using synthetic division on <math>x^3-3x^2-10x+24</math>, we test 2 as a root of this function. We get a remainder of 0. So, we have that 2 is a root of <math> x^3-3x^2-10x+24</math>.<br />
|-<br />
|By synthetic division, <math>x^3-3x^2-10x+24=(x-2)(x^2-x-12)</math>.<br />
|}<br />
<br />
{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"<br />
! Step 4:<br />
|-<br />
|Thus, <math>f(x)=(x-1)(x-2)(x^2-x-12)=(x-1)(x-2)(x-4)(x+3)</math>.<br />
|-<br />
|The zeros of <math>f(x)</math> are <math>1,2,4,-3</math>.<br />
|}<br />
<br />
{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"<br />
! Final Answer:<br />
|-<br />
|The possible roots of <math>f(x)</math> are <math>\pm\{1,2,3,4,6,8,12,24\}</math>.<br />
|-<br />
|The zeros of <math>f(x)</math> are <math>1,2,4,-3</math><br />
|}<br />
<br />
[[004 Sample Final A|<u>'''Return to Sample Exam</u>''']]</div>
MathAdmin