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: (B)

$P ( x )=( x -1)( x +2)( x -3)( x -6)-100 $ $=\left(x^2-4 x+3\right)\left(\underline{x^2-4 x}-12\right)-100$ $=\left(x^2-4 x\right)^2-9\left(x^2-4 x\right)-136 $ $P(x)=\left(x^2-4 x-17\right)\left(x^2-4 x+8\right)$ $ P ( x )=0$ $\therefore \text { equation has two distinct real and two imaginary roots. } $