Question

If \(x=\frac{25}{3-4i}\) then \(2x^{3}-11x^{2}+44x+27=\)

• A. \(-2\)
• B. \(-4\)
• C. \(2\)
• D. \(4\)

Hint:

Simplify x to 3+4i, find the quadratic it satisfies and divide the cubic by it.

Answer 1

The Correct Option is C

Step 1: Reduce powers
From $x^{2}=6x-25$, get $x^{3}=6x^{2}-25x=6(6x-25)-25x=11x-150$.

Step 2: Substitute
$2x^{3}-11x^{2}+44x+27=2(11x-150)-11(6x-25)+44x+27$.

Step 3: Simplify
$=22x-300-66x+275+44x+27=(22-66+44)x+2=2$.

Final Answer:
All x terms cancel, leaving 2. \[ \boxed{\text{(C) }2} \]