4s[sqare]-4s[plus] 1 find the zeroes of given polynomial and find coefficients and its zroes

If this is 4s² - 4s + 1 then:

 

The coefficients of the polynomial are 4, -4 and 1.

 

Because it can be written as (2s - 1)² there are two, i.e. repeated zeros, both at s = 1/2 

 

 

If it is 4s² -4(s+1) then:

 

The polynomial is 4s² - 4s - 4 with coefficients 4, -4 and -4.

 

It doesn't factor so it's zeros are:

 

$${\mathtt{4}}{\mathtt{\,\times\,}}{{\mathtt{s}}}^{{\mathtt{2}}}{\mathtt{\,-\,}}{\mathtt{4}}{\mathtt{\,\times\,}}{\mathtt{s}}{\mathtt{\,-\,}}{\mathtt{4}} = {\mathtt{0}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{s}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{5}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
{\mathtt{s}} = {\frac{\left({\sqrt{{\mathtt{5}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{2}}}}\\
\end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{s}} = -{\mathtt{0.618\: \!033\: \!988\: \!749\: \!894\: \!8}}\\
{\mathtt{s}} = {\mathtt{1.618\: \!033\: \!988\: \!749\: \!894\: \!8}}\\
\end{array} \right\}$$

 

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