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

2015-06-01T01:19:25Z

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 = "te..."

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''' 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
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|1) What does the Rational Zeros Theorem say about the possible zeros?
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|2) How do you check if a possible zero is actually a zero?
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|3) How do you find the rest of the zeros?
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|Answer:
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|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.
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|2) Use synthetic division, or plug a possible zero into the function. If you get 0, you have found a zero.
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|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.
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 1:
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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>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! 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 <math>x^4 + x^3 +2x - 4 = (x - 1)(x^3 +2x^2 + 2x +4)</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 3:
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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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|<math>x^4 +x^3+ 2x -4 = (x - 1)(x + 2)(x^2 + 2)</math>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 4:
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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>
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Final Answer:
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| <math>x^4 + x^3 +2x - 4 = (x - 1)(x + 2)(x - \sqrt{2}i)(x + \sqrt{2}i)</math>
|}

[[005 Sample Final A|'''<u>Return to Sample Exam</u>''']]

005 Sample Final A, Question 6 - Revision history

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

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