question_answer

If three zeroes of a polynomial \[{{x}^{4}}-{{x}^{3}}-3{{x}^{2}}+3x\] are \[0,\sqrt{3}\]  and \[-\sqrt{3}\], then find the fourth zero.

Answer:

Let                    \[P(x)={{x}^{4}}-{{x}^{3}}-3{{x}^{2}}+3x\]

Given, \[0,\sqrt{3},\,\,-\sqrt{3}\] are three zeroes, so

\[x=0\],

\[x=\sqrt{3}\] and \[x=-\sqrt{3}\]

\[\Rightarrow \]               \[(x-\sqrt{3})=0\] and \[x+\sqrt{3}=0\]

Here, \[x(x+\sqrt{3})(x-\sqrt{3})\] will also be the factor of \[P(x)\]

Or, \[x({{x}^{2}}-3)\] will be the factor of \[P(x)\].

then

quotient \[=(x-1)\]

So fourth zero \[\Rightarrow x-1=0\]

\[x=1\]

Hence four zeroes will be \[1,0,\sqrt{3},-\sqrt{3}\].