Final answer:
The equation x^3 - 27 = 0 is of degree 3, which is a polynomial equation. It can be factored as a difference of cubes, leading to a quadratic equation and a linear equation. The quadratic equation has no real solutions, leaving x = 3 as the only real solution.
Step-by-step explanation:
The equation x^3 – 27 = 0 is a polynomial equation, and the first step to solving it is to identify its degree. The degree of a polynomial is the highest power of the variable in the equation. In this case, the highest power of x is 3, so the degree of the polynomial is 3.
To solve for x, we can factor the left side as a difference of cubes because 27 is a cube number (3^3). This gives us:
(x - 3)(x^2 + 3x + 9) = 0
Setting each factor equal to zero gives:
- x - 3 = 0, which solves to x = 3
- x^2 + 3x + 9 = 0, which is a quadratic equation
The quadratic equation can be solved using the quadratic formula, x = [-b ± sqrt(b^2 - 4ac)]/(2a), but in this case, the discriminant (b^2 - 4ac) is negative, indicating there are no real solutions for this part of the equation. So the only real solution to the original equation is x = 3.