Question

State the number and type of roots of the equation 8x^3 - 27 = 0.

a. one positive real, one negative real, and one complex
b. one positive real, two complex
c. one negative real, two complex
d. two positive real, one complex

Answer 1

The equation 8x^3 - 27 = 0 can be solved by factoring the difference of cubes. It has one positive real root and two complex roots.

Step-by-step explanation:

The equation 8x^3 - 27 = 0 can be solved by factoring the difference of cubes. Rearranging the equation, we have (2x)^3 - 3^3 = 0 which can be factored as (2x - 3)(4x^2 + 6x + 9) = 0.

To find the roots, we set each factor equal to zero.

Setting 2x - 3 = 0, we get x = 3/2 which is a positive real root. Setting 4x^2 + 6x + 9 = 0, we can use the quadratic formula to find the remaining two roots.

Since the discriminant is negative, the quadratic has two complex roots. Therefore, the equation has one positive real root and two complex roots.