Algebra Exercise Level Back 1 Science (Leb_Edu_System)

Exercise:

Suppose that $\alpha \,,\beta $ are roots of ${{x}^{2}}-3x+1=0$

What is the value of $E\left( \alpha ,\beta  \right)=\frac{{{\alpha }^{2014}}+{{\beta }^{2014}}+{{\beta }^{2016}}+{{\alpha }^{2016}}}{{{\alpha }^{2015}}+{{\beta }^{2015}}}$

Solution: we have $\alpha \,\And \beta $ are two roots of $p\left( x \right)={{x}^{2}}-3x+1$ hence $p\left( \alpha  \right)=0\,\,\And \,p\left( \beta  \right)=0$

Thus $p\left( \alpha  \right)={{\alpha }^{2}}-3\alpha +1=0$ and $p\left( \beta  \right)={{\beta }^{2}}-3\beta +1=0$

Hence we get $\left( {{\alpha }^{2}},{{\beta }^{2}} \right)=\left( 3\alpha -1,3\beta -1 \right)$

Now back to the $E\left( \alpha ,\beta  \right)=\frac{{{\alpha }^{2014}}\left( 1+{{\alpha }^{2}} \right)+{{\beta }^{2014}}\left( 1+{{\beta }^{2}} \right)}{{{\alpha }^{2015}}+{{\beta }^{2015}}}=\frac{{{\alpha }^{2014}}\left( 3\alpha  \right)+{{\beta }^{2014}}\left( 3\beta  \right)}{{{\alpha }^{2015}}+{{\beta }^{2015}}}=3$