Question

Factor x² with imaginary numbers:

a) (x + i)(x - i)
b) (x - i)(x - i)
c) (x + i)(x + i)
d) (x - i)(x + i)

Answer 1

The expression x² can be factored with imaginary numbers as (x + i)(x - i), which simplifies to x² + 1. Therefore, the correct factoring of x² using imaginary numbers is option d) (x - i)(x + i).

Step-by-step explanation:

The student has asked about factoring x² with imaginary numbers. In general, for any real number a, the expression a² can be factored over the complex numbers as (a + i)(a - i), where i is the imaginary unit, defined as the square root of -1.

Applying this to x², we see that it can be factored as (x + i)(x - i), because:

(x + i)(x - i) = x² - ix + ix - i² = x² + 1 (since i² = -1)

Thus, the correct option that factors x² into a product of two binomials with imaginary numbers is option d) (x - i)(x + i).