Question 149741
a)



Any rational zero can be found through this equation


*[Tex \LARGE Roots=\frac{p}{q}] where p and q are the factors of the last and first coefficients



So let's list the factors of 8 (the last coefficient):


*[Tex \LARGE p=\pm1, \pm2, \pm4, \pm8]


Now let's list the factors of 4 (the first coefficient):


*[Tex \LARGE q=\pm1, \pm2, \pm4]


Now let's divide each factor of the last coefficient by each factor of the first coefficient



*[Tex \LARGE \frac{1}{1}, \frac{1}{2}, \frac{1}{4}, \frac{2}{1}, \frac{2}{2}, \frac{2}{4}, \frac{4}{1}, \frac{4}{2}, \frac{4}{4}, \frac{8}{1}, \frac{8}{2}, \frac{8}{4}, -\frac{1}{1}, -\frac{1}{2}, -\frac{1}{4}, -\frac{2}{1}, -\frac{2}{2}, -\frac{2}{4}, -\frac{4}{1}, -\frac{4}{2}, -\frac{4}{4}, -\frac{8}{1}, -\frac{8}{2}, -\frac{8}{4}]







Now simplify


These are all the distinct rational zeros of the function that could occur


*[Tex \LARGE  1, \frac{1}{2}, \frac{1}{4}, 2, 4, 8, -1, -\frac{1}{2}, -\frac{1}{4}, -2, -4, -8]





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b)



Now let's use synthetic division to test each possible zero:





Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{1/2}}}:

<table cellpadding=10><tr><td>1/2</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>2</td><td>7/2</td><td>21/4</td><td>-115/8</td></tr><tr><td></td><td></td><td>4</td><td>7</td><td>21/2</td><td>-115/4</td><td>-51/8</td></tr></tr></table>

Since the remainder {{{-51/8}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{1/2}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}



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Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{1/4}}}:

<table cellpadding=10><tr><td>1/4</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>1</td><td>3/2</td><td>17/8</td><td>-255/32</td></tr><tr><td></td><td></td><td>4</td><td>6</td><td>17/2</td><td>-255/8</td><td>1/32</td></tr></tr></table>

Since the remainder {{{1/32}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{1/4}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}



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Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{2}}}:

<table cellpadding=10><tr><td>2</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>8</td><td>26</td><td>66</td><td>64</td></tr><tr><td></td><td></td><td>4</td><td>13</td><td>33</td><td>32</td><td>72</td></tr></tr></table>

Since the remainder {{{72}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{2}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}



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Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{4}}}:

<table cellpadding=10><tr><td>4</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>16</td><td>84</td><td>364</td><td>1320</td></tr><tr><td></td><td></td><td>4</td><td>21</td><td>91</td><td>330</td><td>1328</td></tr></tr></table>

Since the remainder {{{1328}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{4}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}



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Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{8}}}:

<table cellpadding=10><tr><td>8</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>32</td><td>296</td><td>2424</td><td>19120</td></tr><tr><td></td><td></td><td>4</td><td>37</td><td>303</td><td>2390</td><td>19128</td></tr></tr></table>

Since the remainder {{{19128}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{8}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}



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Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{-1}}}:

<table cellpadding=10><tr><td>-1</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>-4</td><td>-1</td><td>-6</td><td>40</td></tr><tr><td></td><td></td><td>4</td><td>1</td><td>6</td><td>-40</td><td>48</td></tr></tr></table>

Since the remainder {{{48}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{-1}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}



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Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{-1/2}}}:

<table cellpadding=10><tr><td>-1/2</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>-2</td><td>-3/2</td><td>-11/4</td><td>147/8</td></tr><tr><td></td><td></td><td>4</td><td>3</td><td>11/2</td><td>-147/4</td><td>211/8</td></tr></tr></table>

Since the remainder {{{211/8}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{-1/2}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}



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Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{-1/4}}}:

<table cellpadding=10><tr><td>-1/4</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>-1</td><td>-1</td><td>-3/2</td><td>71/8</td></tr><tr><td></td><td></td><td>4</td><td>4</td><td>6</td><td>-71/2</td><td>135/8</td></tr></tr></table>

Since the remainder {{{135/8}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{-1/4}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}



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Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{-2}}}:

<table cellpadding=10><tr><td>-2</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>-8</td><td>6</td><td>-26</td><td>120</td></tr><tr><td></td><td></td><td>4</td><td>-3</td><td>13</td><td>-60</td><td>128</td></tr></tr></table>

Since the remainder {{{128}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{-2}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}



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Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{-4}}}:

<table cellpadding=10><tr><td>-4</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>-16</td><td>44</td><td>-204</td><td>952</td></tr><tr><td></td><td></td><td>4</td><td>-11</td><td>51</td><td>-238</td><td>960</td></tr></tr></table>

Since the remainder {{{960}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{-4}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}



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Let's make the synthetic division table for the function {{{4x^4+5x^3+7x^2-34x+8}}} given the possible zero {{{-8}}}:

<table cellpadding=10><tr><td>-8</td><td>|</td><td>4</td><td>5</td><td>7</td><td>-34</td><td>8</td></tr><tr><td></td><td>|</td><td> </td><td>-32</td><td>216</td><td>-1784</td><td>14544</td></tr><tr><td></td><td></td><td>4</td><td>-27</td><td>223</td><td>-1818</td><td>14552</td></tr></tr></table>

Since the remainder {{{14552}}} (the right most entry in the last row) is <font size=4><b>not</b></font> equal to zero, this means that {{{-8}}} is <font size=4><b>not</b></font> a zero of {{{4x^4+5x^3+7x^2-34x+8}}}




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Since none of the possible rational roots are actual roots, this means that the polynomial either has irrational roots or complex roots.