Question

lf $f\left(x\right)=x^3+a{x}^2+bx+c=0$ has roots $a,b,c$ and $a,b,c\in R$. If the roots of $x^3+a_1{x}^2+b_{1}x+c_1=0$ are $(\alpha -\beta {)}^2,(\beta -\gamma {)}^2$ and $(\gamma -\alpha {)}^2$, then for $c_1=0$, roots of $f\left(x\right)=0$ are

• A. real and distinct
• B. such that at least two of them are equal
• C. such that two of them are non real
• D. real and equal

Answer 1

Answer: (B)

such that at least two of them are equal

As $(\alpha -\beta {)}^2,(\beta -\gamma {)}^2,(\gamma -\alpha {)}^2$ are roots of $x^3+a_1{x}^2+b_{1}x+c_1=0$
Given
$c_1=0\Rightarrow (\alpha -\beta {)}^2(\beta -\gamma {)}^2(\gamma -\alpha {)}^2=0\Rightarrow (\alpha -\beta \left)\right(\beta -\gamma \left)\right(\gamma -\alpha )=0$
And for this
$\alpha =\beta$ or $\beta =\gamma$ or $\gamma =\alpha$