Question

How many real and complex roots exist for the polynomial x³ −64?

A) 1
B) 2
C) 3
D) 4

Answer 1

C) 3.

The polynomial x³ − 64 has a total of three roots: one real root and two complex roots, found by factoring it as a difference of cubes.

Step-by-step explanation:

The polynomial in question is x³ − 64. This is a difference of cubes, since 64 is 4³. We can factor it using the formula for the difference of cubes, a³ − b³ = (a − b)(a² + ab + b²). This gives us:

• (x − 4)(x² + 4x + 16)

The linear factor (x − 4) provides one real root, x = 4. The quadratic factor (x² + 4x + 16) does not factor further over the reals, as it has no real roots. However, it has two complex roots. Therefore, the polynomial has one real root and two complex roots, totaling three roots.

The correct answer to the question of how many real and complex roots exist for the polynomial x³ − 64 is C) 3.