If $\alpha$ and $\beta$ are zeroes of a quadratic polynomial $4x^{2}+4x+1=0$, then form a quadratic polynomial whose zeroes are $2\alpha$ and $2\beta$.
Given: $\alpha$ and $\beta$ are zeroes of a quadratic polynomial $4x^{2}+4x+1=0$.
To do: To form a quadratic polynomial whose zeroes are $2\alpha$ and $2\beta$.
Solution:
Given polynomial is $4x^{2}+4x+1=0$
$\Rightarrow \frac{4}{4}x^{2}+\frac{4}{4}x+\frac{1}{4}=0$
$\Rightarrow x^{2}+x+\frac{1}{4}=0$
$\Rightarrow \alpha+\beta=1\ and\ \alpha.\beta=\frac{1}{4}$
If there are $2\alpha\ and\ 2\beta$ two zeroes of a polynomial, then
Sum of the zeroes$=2\alpha+2\beta=2( \alpha+\beta)=2\times1=2$
Product of the zeroes$=2\alpha.2\beta=4( \alpha.\beta)=4\times\frac{1}{4}=1$
Thus the polynomial with $2\alpha\ and\ 2\beta$ is:
$x^{2}+( Sum\ of\ the\ zeroes)x+( Product\ of\ the\ zeroes)=0$
$x^{2}+2x+1=0$.
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