Final answer:
The Irrational Root Theorem states that irrational roots of a polynomial with integer coefficients must occur in conjugate pairs, like the roots √2 and -√2 of the equation x^2 - 2 = 0.
Step-by-step explanation:
Irrational Root Theorem
The Irrational Root Theorem is a mathematical principle that applies to polynomial equations, particularly stating that any irrational roots (non-repeating, non-terminating decimals that cannot be written as a simple fraction) of a polynomial equation with integer coefficients must occur in conjugate pairs. This means if a + b√(c) is an irrational root, where a, b, and c are rational numbers and c is not a perfect square, then its conjugate a - b√(c) will also be a root of the polynomial.
For example, consider the polynomial equation x^2 - 2 = 0. Its roots are √2 and -√2, which are both irrational and are conjugates of each other. Another example is the polynomial x^2 - 3x + 1. If it has an irrational root of the form a + b√(c), there must be another root a - b√(c) in the equation, ensuring that when multiplied, the result is rational.
The Irrational Root Theorem is particularly useful in identifying possible irrational solutions to polynomial equations and understanding the symmetry in the roots that occur due to this theorem.