Question

The equation whose roots are the squares of the roots of $x^3+ax+b=0$, is

• A. $x^3+2a{x}^2-a^{2}x-b^2=0$
• B. $x^3+2a{x}^2+a^{2}x-b^2=0$
• C. $x^6+a{x}^2+6=0$
• D. $x^6-a{x}^2+6=0$

Answer 1

The correct option is B $x^3+2a{x}^2+a^{2}x-b^2=0$

Let $y=x^2$
$x^3+ax+b=0\Rightarrow x^3+ax=-b\Rightarrow {(x^3+ax)}^2={(-b)}^2\Rightarrow x^6+a^2{x}^2+2a{x}^4=b^2$
Substituting $y=x^2$, we get
$y^3+a^{2}y+2a{y}^2=b^2\Rightarrow y^3+2a{y}^2+a^{2}y-b^2=0$
Now replacing $y$ by $x$
$x^3+2a{x}^2+a^{2}x-b^2=0$ is the eqaution whose roots are square of roots of $x^3+ax+b=0$