if $\alpha$ and $\beta$ are zeroes of polynomial $x^{2}-2x-15$, then form a quadratic polynomial whose zeroes are $2\alpha$ and $2\beta$.

Given: $\alpha$ and $\beta$ are zeroes of polynomial $x^{2}-2x-15$.

To do: To form a quadratic polynomial whose zeroes are $2\alpha$ and $2\beta$.

Solution:

As given,  $x^{2}-2x-15$

$=x^2-5x+3x-15$

$=x( x-5)+3( x-5)$

$=( x+3)( x-5)$

Let $\alpha=-3$ and $\beta=5$

$\alpha+\beta=-3+5=2$ and $\alpha\beta=-3\times\ 5=-15$

$\therefore$ The polynomial whose zeroes are $2\alpha$ and $2\beta$

$=k( x^2-( 2\alpha+2\beta)x+2\alpha2\beta)$

$=k( x^2-2( \alpha+\beta)x+4\alpha\beta)$

$=k( x^2-2\times( -2)x+4\times( -15))$

$=k( x^2+4x-60)$