Difference between revisions of "004 Sample Final A, Problem 8"

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(Created page with "<span class="exam"> a) List all the possible zeros of the function.<br> b) Find all the zeros, that is, solve <math>f(x) = 0</math>")
 
 
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<span class="exam"> a) List all the possible zeros of the function.<br>
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<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</math>
 
b) Find all the zeros, that is, solve <math>f(x) = 0</math>
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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! Foundations
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|-
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|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>?
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|-
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|Answer:
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|-
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|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>.
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|}
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Solution:
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{| class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 1:
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|-
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|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>.
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|}
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 2:
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|-
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|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>.
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|-
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|By synthetic division, <math>f(x)=(x-1)(x^3-3x^2-10x+24)</math>.
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|}
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 3:
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|-
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|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>.
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|-
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|By synthetic division, <math>x^3-3x^2-10x+24=(x-2)(x^2-x-12)</math>.
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|}
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Step 4:
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|-
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|Thus, <math>f(x)=(x-1)(x-2)(x^2-x-12)=(x-1)(x-2)(x-4)(x+3)</math>.
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|-
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|The zeros of <math>f(x)</math> are <math>1,2,4,-3</math>.
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|}
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{|class = "mw-collapsible mw-collapsed" style = "text-align:left;"
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! Final Answer:
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|-
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|The possible roots of <math>f(x)</math> are <math>\pm\{1,2,3,4,6,8,12,24\}</math>.
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|-
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|The zeros of <math>f(x)</math> are <math>1,2,4,-3</math>
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|}
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[[004 Sample Final A|<u>'''Return to Sample Exam</u>''']]

Latest revision as of 17:11, 4 May 2015

a) List all the possible rational zeros of the function
b) Find all the zeros, that is, solve

Foundations
If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^4+bx^3+cx^2+dx+e} , what does the rational roots tell us are the possible roots of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} ?
Answer:
The rational roots tells us that the possible roots of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm k} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is a divisor of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e} .


Solution:

Step 1:
By the rational roots test, the possible roots of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\{1,2,3,4,6,8,12,24\}} .
Step 2:
Using synthetic division, we test 1 as a root of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} . We get a remainder of 0. So, we have that 1 is a root of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} .
By synthetic division, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=(x-1)(x^3-3x^2-10x+24)} .
Step 3:
Using synthetic division on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3-3x^2-10x+24} , we test 2 as a root of this function. We get a remainder of 0. So, we have that 2 is a root of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3-3x^2-10x+24} .
By synthetic division, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^3-3x^2-10x+24=(x-2)(x^2-x-12)} .
Step 4:
Thus, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=(x-1)(x-2)(x^2-x-12)=(x-1)(x-2)(x-4)(x+3)} .
The zeros of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,2,4,-3} .
Final Answer:
The possible roots of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pm\{1,2,3,4,6,8,12,24\}} .
The zeros of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1,2,4,-3}

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