Question

How many distinct real roots (solutions) does the equation x⁴ - x = 0 have?

1) 0
2) 1
3) 2
4) 3
5) 4

Answer 1

The equation x⁴ - x = 0 has two distinct real roots: x = 0 and x = 1, as the cubic polynomial can be factored into x(x - 1)(x² + x + 1) = 0, but x² + x + 1 = 0 does not have any real solutions.

Step-by-step explanation:

To determine how many distinct real roots the equation x4 - x = 0 has, we can factor the equation. Factoring out an x, we get x(x3 - 1) = 0. This can further be factored into x(x - 1)(x2 + x + 1) = 0. By the Zero Product Property, this equation has roots when x = 0, x = 1, and when x2 + x + 1 = 0.

The quadratic x2 + x + 1 = 0 does not have any real roots because its discriminant (b2 - 4ac) is negative. Therefore, the original equation has two distinct real roots: x = 0 and x = 1.

The ability to factor polynomials is essential in solving such algebraic equations.