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

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

Tardigrade Admin · Accepted Answer

p ( x )=( x -1)( x +2)( x -3)( x -6)-100 =( x 2-4 x +3)( x 2-4 x -12)-100 p ( x )=( x 2-4 x +3)2-15( x 2-4 x +3)-100 ⇒ x 2-4 x +3=20 text and x 2-4 x +3=-5 ⇒ p ( x )=0 text has two distinct real roots.

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

all four distinct real roots

three distinct real roots

two real and two imaginary roots

all four imaginary roots

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

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