Q. The polynomial $\left( ax^{2} + bx + c \right)\left( ax^{2} - dx - c \right),\ ac \neq 0$ has
A) Four real roots
B) At least two real roots
C) At most two real roots
D) No real roots
Explanation
(b)
Let $f(x) = \left( ax^{2} + bx + c \right)\left( ax^{2} - dx - c \right)$
$\Rightarrow D_{1} = b^{2} - 4ac\$and$\ D_{2} = d^{2} + 4ac$
$\Rightarrow D_{1} + D_{2} = b^{2} - 4ac + d^{2} + 4ac$
$= b^{2} + d^{2}\ \geq 0$
$\therefore$ At least one of $D_{1}$and $D_{2}$ is positive
Hence, the polynomial has at least two real roots
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.