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 16:11, 4 May 2015

a) List all the possible rational zeros of the function 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^4-4x^3-7x^2+34x-24.}
b) Find all the zeros, that is, solve 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) = 0}

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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