Question

A real number x₀ is considered algebraic 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 transcendental equation
d) Is a solution to a linear equation

Answer 1

A real number x₀ is considered algebraic if it is a solution to a polynomial equation with integer coefficients, such as a quadratic equation ax²+bx+c = 0. Algebraic numbers can be rational if they are also expressible as a ratio of two integers, but not all algebraic numbers are rational.

Step-by-step explanation:

A real number x₀ is considered algebraic if it is a solution to a polynomial equation with integer coefficients. This means that algebraic numbers include those that solve equations such as ax²+bx+c = 0, where a, b, and c are integers. For example, the solutions to the quadratic equation x²+1.2 x 10⁻²x -6.0 × 10⁻³ = 0 can be found using the quadratic formula and if the solutions are real, they are considered algebraic numbers.

However, if a number can be expressed as a ratio of two integers, it is not just algebraic but also rational. Not all algebraic numbers are rational. Transcendental numbers, such as π and e, are not algebraic because they are not solutions to any polynomial equation with integer coefficients. Also, being a solution to a linear equation doesn't necessarily make a number algebraic unless the coefficients are integers.