Difference between revisions of "005 Sample Final A, Question 6"

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(Created page with "''' Question ''' Factor the following polynomial completely,     <math>p(x) = x^4 + x^3 + 2x-4 </math> {| class="mw-collapsible mw-collapsed" style = "...")
 
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! Final Answers
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! Step 1:
 
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|a) False. Nothing in the definition of a geometric sequence requires the common ratio to be always positive. For example, <math>a_n = (-a)^n</math>
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| First, we use the Rational Zeros Theorem to note that the possible zeros are: <math>\{\pm 1, \pm 2, \pm 4 \}</math>
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! Step 2:
 
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|b) False. Linear systems only have a solution if the lines intersect. So y = x and y = x + 1 will never intersect because they are parallel.
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| Now we start checking which of the possible roots are actually roots. We find that 1 is a zero, and apply either synthetic division or long division to get <math>x^4 + x^3 +2x - 4 = (x - 1)(x^3 +2x^2 + 2x +4)</math>
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! Step 3:
 
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|c) False. <math>y = x^2</math> does not have an inverse.
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| We continue checking zeros and find that -2 is a zero. Applying synthetic division or long division we can simplify the polynomial down to:
 
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|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
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|<math>x^4 +x^3+ 2x -4 = (x - 1)(x + 2)(x^2 + 2)</math>
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! Step 4:
 
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|e) True.
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| Now we can finish the problem  by applying the quadratic formula or just finding the roots of <math>x^2 + 2</math>
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! Final Answer:
 
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|f) False.
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| <math>x^4 + x^3 +2x - 4 = (x - 1)(x + 2)(x - \sqrt{2}i)(x + \sqrt{2}i)</math>
 
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Revision as of 14:13, 17 May 2015

Question Factor the following polynomial completely,     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 p(x) = x^4 + x^3 + 2x-4 }


Step 1:
First, we use the Rational Zeros Theorem to note that the possible zeros 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, \pm 2, \pm 4 \}}
Step 2:
Now we start checking which of the possible roots are actually roots. We find that 1 is a zero, and apply either synthetic division or long division to get 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^4 + x^3 +2x - 4 = (x - 1)(x^3 +2x^2 + 2x +4)}
Step 3:
We continue checking zeros and find that -2 is a zero. Applying synthetic division or long division we can simplify the polynomial down to:
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^4 +x^3+ 2x -4 = (x - 1)(x + 2)(x^2 + 2)}
Step 4:
Now we can finish the problem by applying the quadratic formula or just finding the 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 x^2 + 2}
Final Answer:
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^4 + x^3 +2x - 4 = (x - 1)(x + 2)(x - \sqrt{2}i)(x + \sqrt{2}i)}
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