Question

If $\alpha$ and $\beta$ are the zeroes of the quadratic polynomial $4x^2+4x+1$, then form a quadratic polynomial whose zeroes are $2\alpha$ and $2\beta$.

• A. $x^2-2x+1$
• B. $x^2+2x+1$
• C. $x^2+x+1$
• D. $x^2+12x+1$

Answer 1

Answer: (B)

$x^2+2x+1$

Given that,

$\alpha$ and $\beta$ are Zeroes, then equation will be

$x^2-\left(\alpha +\beta \right)+\alpha \beta =0$

Given equation is $4x^2+4x+1=0$

$\begin{array}{c}\therefore \alpha +\beta =\frac{-4}{4}=-1\\ \alpha \beta =\frac{1}{4}\end{array}$

Now, new zeroes are $2\alpha$ and $2\beta$

then,

$\begin{array}{c}2\alpha +2\beta =2\left(\alpha +\beta \right)=2\left(-1\right)=-2\\ 2\alpha \times 2\beta =2\alpha \beta =4\times \left(\frac{1}{4}\right)=1\end{array}$

So the quadratic polynomial will be

$\begin{array}{c}{x}^2-\left(-2\right)x-1=0\\ \Rightarrow x^2+2x+1=0\end{array}$

Option $B$ is correct answer.