Question

If α and β are the zeroes of the polynomial $f\left(x\right)=x^2+ax+b,$ then the polynomial whose zeroes are ${\alpha }^2+{\beta }^2+2\alpha \beta and{\alpha }^2+{\beta }^2-2\alpha \beta is$

• A. $k[x^2-3(a^2+b)$x + ab - 4}
• B. $k[x^2-2(a^2-2b)x+a^2(a^2-4b)$ }
• C. $k[x^2-(a-2b)x+ab(a-2)$}
• D. $k[x^2-abx+a^2(a+b\left)\right]$}

Answer 1

The correct option is B $k[x^2-2(a^2-2b)x+a^2(a^2-4b)$ }

Consider, $f\left(x\right)=x^2+ax+b$

Sum of zeroes, $\alpha +\beta =-a$

Product of zeroes, $\alpha \beta =b$

So, ${\alpha }^2+{\beta }^2+2\alpha \beta =(\alpha +\beta {)}^2=(-a{)}^2=a^2$

${\alpha }^2+{\beta }^2-2\alpha \beta =(\alpha +\beta {)}^2-4\alpha \beta =(-a{)}^2-4b=a^2-4b$

Now, the zeroes of the required quadratic polynomial are ${\alpha }^2+{\beta }^2+2\alpha \beta and{\alpha }^2+{\beta }^2-2\alpha \beta$

Sum of zeroes $=\left(a^2\right)+(a^2-4b)=2a^2-4b=2(a^2-2b)$

Product of zeroes $=a^2(a^2-4b)$

∴ Required polynomial $=k[x^2-2(a^2-2b)x+a^2(a^2-4b\left)\right]$

Hence, the correct answer is option (b).