Question

Princess Poly claims that the equation (3x−4i)(7x−8)(3x+4i)=0 has three roots, a real root and a complex root with a multiplicity of 2.
Which statement regarding her claim is true?


A) Princess Poly is correct. The equation has exactly three roots, one real root, x=87, and one complex root with a multiplicity of 2, x=43i.

B) Princess Poly is not correct. The equation has exactly three roots, one real root, x=−87, and two complex roots, x=43i and x=−43i.

C) Princess Poly is not correct. The equation has exactly three roots, one real root, x=87, and two complex roots, x=43i and x=−43i.

D) Princess Poly is correct. The equation has exactly three roots, one real root, x=87, and one complex root with a multiplicity of 2, x=−43i.

Answer 1

Given equation:

$(3x-4i)(7x-8)(3x+4i)=0$

i.e., $3x-4i=0$       and       $7x-8=0$       and       $3x+4i=0$

i.e., $3x=4i$       and       $7x=8$       and       $3x=-4i$

i.e., $x=\frac{4i}{3}$       and       $x=\frac{8}{7}$       and       $x=-\frac{4i}{3}$

Therefore, the given equation has a real root i.e., $x=\frac{8}{7}$

and two complex conjugate roots i.e., $x=\frac{4i}{3}$ and $x=-\frac{4i}{3}$

Hence, option C is correct.