If $\alpha$ and $\beta$ are the zeroes of a polynomial such that $\alpha+\beta=-6$ and $\alpha\beta=5$, then find the polynomial.

Given: $\alpha$ and $\beta$ are the zeroes of a polynomial such that $\alpha+\beta=-6$ and $\alpha\beta=5$.

To do: To find the polynomial.

Solution:

The polynomial is:

$p( x)=x^2-( sum\ of\ the\ zeroes)x+( product\ of\ the\ zeroes)$

$p( x)=x^2-( \alpha+\beta)x+\alpha\beta$

$p( x)=x^2-( -6)x+( 5)$

$p( x)=x^2+6x+5$