Question

Consider the polynomial $p ( x )=( x -1)( x +2)( x -3)( x -6)-100$. The equation $p ( x )=0$ has

• A. all four distinct real roots
• B. three distinct real roots
• C. two real and two imaginary roots
• D. all four imaginary roots

Answer 1

Answer: (C)

$ p ( x )=( x -1)( x +2)( x -3)( x -6)-100$
$=\left( x ^2-4 x +3\right)\left( x ^2-4 x -12\right)-100 $
$p ( x )=\left( x ^2-4 x +3\right)^2-15\left( x ^2-4 x +3\right)-100$
$\Rightarrow x ^2-4 x +3=20 \text { and } x ^2-4 x +3=-5$
$\Rightarrow p ( x )=0 \text { has two distinct real roots. }$