Fundamental Test of Algebra

Algebra Level 4

If $1024(11-5ix) = 11x^{10} + 20ix^9$ for $i = \sqrt{-1}$ , then which of the following is true regarding the magnitude of $x$ ?

9 possible positive real values less than 10 None of the others 10 possible positive real values less than 11 9 possible positive rational values

1 solution

Tom Engelsman
Oct 19, 2017

Matching the real and the imaginary components of this equation produces:

REAL: $11,264 = 11x^{10} \Rightarrow x = \pm2$

IMAGINARY: $-5,120x = 20x^9 \Rightarrow x(20x^8 + 5120) = 0 \Rightarrow x = 0, \pm2i$

There is no $x \in \mathbb{C}$ that solves this equation $\Rightarrow$ NONE OF THESE CHOICES.

@Tom Engelsman If $x$ is a complex number, how can you directly compare the real and imaginary parts?

Satvik Golechha - 3 years, 5 months ago