Question

A real number x₀ is termed transcendental if it:

a) Is a solution to a polynomial equation with integer coefficients
b) Can be expressed as a ratio of two integers
c) Is a solution to a linear equation
d) Is not a solution to any polynomial equation with integer coefficients

Answer 1

A transcendental number is not a solution to any polynomial equation with integer coefficients, unlike algebraic numbers which are solutions to such polynomial equations.

Step-by-step explanation:

A transcendental number is a real number that is not a solution to any polynomial equation with integer coefficients. The correct answer to the question posed is d) Is not a solution to any polynomial equation with integer coefficients. This means that transcendental numbers cannot be expressed as the roots of any non-zero polynomial equation such as ax2+bx+c = 0 where a, b, and c are integers and a is not zero.

An example of a transcendental number is π (pi), as there is no polynomial with integer coefficients for which π is a solution. Another well-known transcendental number is e, the base of the natural logarithms. In contrast to transcendental numbers, algebraic numbers are those that are solutions to polynomial equations with integer coefficients, such as the roots of the quadratic equation ax2+bx+c = 0.