005 Sample Final A, Question 6 - Revision history

Matt Lee at 03:19, 22 May 2015

2015-05-22T03:19:33Z

Matt Lee at 22:13, 17 May 2015

2015-05-17T22:13:00Z

Matt Lee: Created page with "''' Question ''' Factor the following polynomial completely, $p(x) = x^4 + x^3 + 2x-4$ {| class="mw-collapsible mw-collapsed" style = "..."

2015-05-01T04:25:22Z

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 = "..."

New page

''' Question ''' Factor the following polynomial completely,     <math>p(x) = x^4 + x^3 + 2x-4 </math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answers
|-
|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>
|-
|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.
|-
|c) False. <math>y = x^2</math> does not have an inverse.
|-
|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
|-
|e) True.
|-
|f) False.
|}

← Older revision		Revision as of 03:19, 22 May 2015
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	''' Question ''' Factor the following polynomial completely,     <math>p(x) = x^4 + x^3 + 2x-4 </math>		''' Question ''' Factor the following polynomial completely,     <math>p(x) = x^4 + x^3 + 2x-4 </math>

		+	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"
		+	!Foundations
		+	\|-
		+	\|1) What does the Rational Zeros Theorem say about the possible zeros?
		+	\|-
		+	\|2) How do you check if a possible zero is actually a zero?
		+	\|-
		+	\|3) How do you find the rest of the zeros?
		+	\|-
		+	\|Answer:
		+	\|-
		+	\|1) The possible divisors can be found by finding the factors of -10, in a list, and the factors of 1, in a second list. Then write down all the fractions with numerators from the first list and denominators from the second list.
		+	\|-
		+	\|2) Use synthetic division, or plug a possible zero into the function. If you get 0, you have found a zero.
		+	\|-
		+	\|3) After your reduce the polynomial with synthetic division, try and find another zero from the list you made in part a). Once you reach a degree 2 polynomial you can finish the problem with the quadratic formula.
		+	\|}

@@ Line 4: / Line 4: @@
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
-! Final Answers
+! Step 1:
 |-
-|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>
+| First, we use the Rational Zeros Theorem to note that the possible zeros are: <math>\{\pm 1, \pm 2, \pm 4 \}</math>
 |-
-|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.
+| 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>
 |-
-|c) False. <math>y = x^2</math> does not have an inverse.
+| We continue checking zeros and find that -2 is a zero. Applying synthetic division or long division we can simplify the polynomial down to:
 |-
-|d) True. <math>cos^2(x) - cos(x) = 0</math> has multiple solutions.
+|<math>x^4 +x^3+ 2x -4 = (x - 1)(x + 2)(x^2 + 2)</math>
 |-
-|e) True.
+| Now we can finish the problem  by applying the quadratic formula or just finding the roots of <math>x^2 + 2</math>
 |-
-|f) False.
+| <math>x^4 + x^3 +2x - 4 = (x - 1)(x + 2)(x - \sqrt{2}i)(x + \sqrt{2}i)</math>
 |}

005 Sample Final A, Question 6 - Revision history

Matt Lee at 03:19, 22 May 2015

Matt Lee at 22:13, 17 May 2015

Matt Lee: Created page with "''' Question ''' Factor the following polynomial completely, p(x) = x^4 + x^3 + 2x-4 {| class="mw-collapsible mw-collapsed" style = "..."

Matt Lee: Created page with "''' Question ''' Factor the following polynomial completely, $p(x) = x^4 + x^3 + 2x-4$ {| class="mw-collapsible mw-collapsed" style = "..."