Question

If a, ß and y are the zeroes of the polynomial x3 + x2 - 2x + 3, then the value
of a2 + B2 + y2 is equal to​

Answer 1

SOLUTION

GIVEN

$\sf{ \alpha , \beta , \gamma }$ are the zeroes of the polynomial

$\sf{ {x}^{3} + {x}^{2} - 2x + 3 = 0 }$

TO DETERMINE

The value of

$\sf{ { \alpha }^{2} + { \beta }^{2} + { \gamma }^{2} }$

EVALUATION

Here it is given that

$\sf{ \alpha , \beta , \gamma }$ are the zeroes of the polynomial

$\sf{ {x}^{3} + {x}^{2} - 2x + 3 = 0 }$

Therefore

$\sf{ \alpha + \beta + \gamma = - 1 }$

$\sf{ \alpha \beta + \alpha \gamma + \beta \gamma = - 2}$

$\sf{ \alpha \beta \gamma = - 3 }$

Now

$\sf{ { \alpha }^{2} + { \beta }^{2} + { \gamma }^{2} }$

$\sf{ = {( \alpha + \beta + \gamma )}^{2} - 2( \alpha \beta + \alpha \gamma + \beta \gamma ) }$

$\sf{ = {( - 1)}^{2} - 2 \times ( - 2 ) }$

$\sf{ = 1 + 4 }$

$\sf{ = 5}$

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