Question 183030
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 -4 (the last coefficient):


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


Now let's list the factors of 1 (the first coefficient):


*[Tex \LARGE q=\pm1]


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



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







Now simplify


These are all the distinct rational zeros of the function that <i>could</i> occur (ie some of these values are NOT zeros, but could be)


*[Tex \LARGE  1, 2, 4, -1, -2, -4]




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Let's see if the possible zero {{{1}}} is really a root for the function {{{x^4-2x^3-3x^2+8x-4}}}



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

<table cellpadding=10><tr><td>1</td><td>|</td><td>1</td><td>-2</td><td>-3</td><td>8</td><td>-4</td></tr><tr><td></td><td>|</td><td> </td><td>1</td><td>-1</td><td>-4</td><td>4</td></tr><tr><td></td><td></td><td>1</td><td>-1</td><td>-4</td><td>4</td><td>0</td></tr></tr></table>

Since the remainder {{{0}}} (the right most entry in the last row) is equal to zero, this means that {{{1}}} is a zero of {{{x^4-2x^3-3x^2+8x-4}}}



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Let's see if the possible zero {{{2}}} is really a root for the function {{{x^4-2x^3-3x^2+8x-4}}}



So let's make the synthetic division table for the function {{{x^4-2x^3-3x^2+8x-4}}} given the possible zero {{{2}}}:

<table cellpadding=10><tr><td>2</td><td>|</td><td>1</td><td>-2</td><td>-3</td><td>8</td><td>-4</td></tr><tr><td></td><td>|</td><td> </td><td>2</td><td>0</td><td>-6</td><td>4</td></tr><tr><td></td><td></td><td>1</td><td>0</td><td>-3</td><td>2</td><td>0</td></tr></tr></table>

Since the remainder {{{0}}} (the right most entry in the last row) is equal to zero, this means that {{{2}}} is a zero of {{{x^4-2x^3-3x^2+8x-4}}}



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Let's see if the possible zero {{{4}}} is really a root for the function {{{x^4-2x^3-3x^2+8x-4}}}



So let's make the synthetic division table for the function {{{x^4-2x^3-3x^2+8x-4}}} given the possible zero {{{4}}}:

<table cellpadding=10><tr><td>4</td><td>|</td><td>1</td><td>-2</td><td>-3</td><td>8</td><td>-4</td></tr><tr><td></td><td>|</td><td> </td><td>4</td><td>8</td><td>20</td><td>112</td></tr><tr><td></td><td></td><td>1</td><td>2</td><td>5</td><td>28</td><td>108</td></tr></tr></table>

Since the remainder {{{108}}} (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 {{{x^4-2x^3-3x^2+8x-4}}}



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Let's see if the possible zero {{{-1}}} is really a root for the function {{{x^4-2x^3-3x^2+8x-4}}}



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

<table cellpadding=10><tr><td>-1</td><td>|</td><td>1</td><td>-2</td><td>-3</td><td>8</td><td>-4</td></tr><tr><td></td><td>|</td><td> </td><td>-1</td><td>3</td><td>0</td><td>-8</td></tr><tr><td></td><td></td><td>1</td><td>-3</td><td>0</td><td>8</td><td>-12</td></tr></tr></table>

Since the remainder {{{-12}}} (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 {{{x^4-2x^3-3x^2+8x-4}}}



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Let's see if the possible zero {{{-2}}} is really a root for the function {{{x^4-2x^3-3x^2+8x-4}}}



So let's make the synthetic division table for the function {{{x^4-2x^3-3x^2+8x-4}}} given the possible zero {{{-2}}}:

<table cellpadding=10><tr><td>-2</td><td>|</td><td>1</td><td>-2</td><td>-3</td><td>8</td><td>-4</td></tr><tr><td></td><td>|</td><td> </td><td>-2</td><td>8</td><td>-10</td><td>4</td></tr><tr><td></td><td></td><td>1</td><td>-4</td><td>5</td><td>-2</td><td>0</td></tr></tr></table>

Since the remainder {{{0}}} (the right most entry in the last row) is equal to zero, this means that {{{-2}}} is a zero of {{{x^4-2x^3-3x^2+8x-4}}}



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Let's see if the possible zero {{{-4}}} is really a root for the function {{{x^4-2x^3-3x^2+8x-4}}}



So let's make the synthetic division table for the function {{{x^4-2x^3-3x^2+8x-4}}} given the possible zero {{{-4}}}:

<table cellpadding=10><tr><td>-4</td><td>|</td><td>1</td><td>-2</td><td>-3</td><td>8</td><td>-4</td></tr><tr><td></td><td>|</td><td> </td><td>-4</td><td>24</td><td>-84</td><td>304</td></tr><tr><td></td><td></td><td>1</td><td>-6</td><td>21</td><td>-76</td><td>300</td></tr></tr></table>

Since the remainder {{{300}}} (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 {{{x^4-2x^3-3x^2+8x-4}}}




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Answer:


So the rational zeros of {{{p(x)=x^4-2x^3-3x^2+8x-4}}} are: 1,2,-2


In other words, if we plug in {{{x=1}}}, {{{x=2}}} or {{{x=-2}}} into {{{p(x)=x^4-2x^3-3x^2+8x-4}}}, we'll get 0 as a result (try it out if you aren't sure)


Note: the zero 1 has a multiplicity of 2 (ie it is counted twice)