Question

Let $p\left(x\right)=x^2+ax+b$ have two distinct real roots, where $a,b$ are real numbers. Define $g\left(x\right)=p\left(x^3\right)$ for all real numbers $x$. Then which of the following statements are true?
I. $g$ has exactly two distinct real roots
II. $g$ can have more than two distinct real roots
III. There exists a real number $\alpha$ such that $g\left(x\right)\ge \alpha$ for all real $x$.

• A. Only I
• B. Only I and III
• C. Only II
• D. Only II and III

Answer 1

The correct option is B Only I and III

$P\left(X\right)=x^2+ax+b$ $a,b\in R$

Roots are real

$g\left(x\right)=p\left(x^3\right)4$

$x^3=t$

$g\left(x\right)=p\left(t\right)$

$p\left(t\right)=t^2+at+b$

Roots are real but $t=x^3$

Let $t=\beta$

$x^3=\beta$

$x=(\beta {)}^{\frac{1}{3}},({\beta }^{\frac{1}{3}})w,({\beta }^{\frac{1}{3}}),w^2$

Only $1$ real root, $2-$image.

Same are other roots

$t=\gamma$

It has only one real root

Total $2-$ distinct real roots and $4-$ Image.