Zeroes are interesting !!!

Algebra Level pending

I f α , β a r e z e r o e s o f x 2 − 3 x + 1 = 0 t h e n f i n d t h e v a l u e o f α 2014 + β 2014 + α 2016 + β 2016 α 2015 + β 2015 If\quad \alpha ,\quad \beta \quad are\quad zeroes\quad of\quad { x }^{ 2 }-3x+1=0\quad then\quad find\quad the\quad value\quad of\quad \\ \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \frac { { \alpha }^{ 2014 }+{ \beta }^{ 2014 }+{ \alpha }^{ 2016 }+{ \beta }^{ 2016 } }{ { \alpha }^{ 2015 }+{ \beta }^{ 2015 } }


The answer is 3.

Aryan Saxena by aryan saxena , 20, India

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2 solutions

Damiann Mangan
Apr 15, 2015

By simplifying the given quadratic equation, we get x 2016 + x 2014 = 3 x 2015 x^{2016} + x^{2014} = 3x^{2015} .

Thus, The problem becomes 3 ( α 2015 + β 2015 ) α 2015 + β 2015 \frac{3(\alpha^{2015} + \beta^{2015})}{\alpha^{2015} + \beta^{2015}} which, easily, is 3 3 .

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Utkarsh Dwivedi
Apr 17, 2015

aryan saxena Don't write the full question , whenever you write , from starting to beginning in '\ [ ' and '\ ]' instead use it only for expressing the mathematical representations.

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