008A Sample Final A, Question 9 - Revision history

MathAdmin at 06:55, 26 May 2015

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MathAdmin at 19:48, 23 May 2015

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MathAdmin at 19:47, 23 May 2015

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MathAdmin at 19:37, 23 May 2015

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MathAdmin: Created page with "'''Question: ''' a) List all the possible rational zeros of the function $f(x) = x^4 + 5x^3 - 27x^2 +31x -10$
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2015-04-28T21:10:28Z

Created page with "'''Question: ''' a) List all the possible rational zeros of the function <math>f(x) = x^4 + 5x^3 - 27x^2 +31x -10</math><br> &..."

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'''Question: ''' a) List all the possible rational zeros of the function <math>f(x) = x^4 + 5x^3 - 27x^2 +31x -10</math><br>
                 
b) Find all the zeros, that is, solve f(x) = 0

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!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.
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Solution:

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
! Step 1:
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|Start by factoring -10, and 1. Then the Rational Zeros Theorem gives us that the possible rational zeros are <math>\pm 1, \pm 2, \pm 5,</math> and <math>\pm 10</math>.
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! Step 2:
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|Start testing zeros with 1 and -1 since they require the least arithmetic. You will also find that 1 is a zero. Applying synthetic division you can reduce the polynomial to <math>x^3 + 6x^2 - 21x + 10</math>.
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! Step 3:
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|Now we just need to find the zeros of <math>x^3 + 6x^2 - 21x + 10</math>. Since we are not down to a quadratic polynomial we have to continue finding zeros from the list of rational zeros we found in step 1. You will find 2 is another root, and the polynomial can further be reduced to <math>x^2 + 8x - 5</math>
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! Step 4:
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|Now that the polynomial has been reduced to a quadratic polynomial we can use the quadratic formula to find the rest of the zeros. By doing so we find the roots are <math>\frac{-8 \pm \sqrt{64 + 20}}{2} = \frac{-8 \pm \sqrt{4*21}}{2} = -4 \pm \sqrt{21}</math>. Thus the zeros of <math>x^4 + 5x^3 - 27x^2 + 31x - 10</math> are <math>1, 2, </math>and <math>-4 \pm \sqrt{21}</math>
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! Final Answer:
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|The roots are <math> x = 1, 2, </math>and <math>-4 \pm \sqrt{21}</math>
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