Question

The polynomial $(a{x}^2+bx+c)(a{x}^2-dx-c),ac\ne 0,$ has

• A. four real zeros
• B. at least two real zeros
• C. at most two real zeros
• D. no real zeros

Answer 1

Answer: (B)

at least two real zeros

For
$a{x}^2+bx+c=0$
$b^2-4ac\ge 0$ for real roots ...(i)
and for
$a{x}^2-dx-c=0$
$d^2+4ac\ge 0$ for real roots ...(ii)
Now,
$ac\ne 0$
Hence,
Case I
If $ac>0$
Hence,
$a{x}^2-dx-x=0$ has positive roots.
Case II
If $ac<0$
Then,
$a{x}^2+bx+c=0$
has Real roots.
Hence, the above polynomial has atleast two real roots.