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 15:13, 17 May 2015

Question Factor the following polynomial completely,     


Step 1:
First, we use the Rational Zeros Theorem to note that the possible zeros are:
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
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:
Step 4:
Now we can finish the problem by applying the quadratic formula or just finding the roots of
Final Answer:
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