The Irrational Root Theorem: Definition & Application

Instructor: Russell Frith
This lesson describes the irrational root theorem and how it may be used to find additional roots for a polynomial. The irrational root theorem works only if the coefficients of the polynomial are rational.

What Is the Irrational Root Theorem?

A polynomial with integer coefficients has the following roots:


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Can you find at least two additional roots? The answer is yes, and the technique used to find them makes use of the irrational root theorem.

The irrational root theorem may be stated as follows:

Let a and b be two numbers such that a is a rational number and the square root of b is an irrational number. The irrational root theorem states that if the irrational sum of a plus the square root of b is the root of a polynomial with rational coefficients, then a minus the square root of b, which is also an irrational number, is also a root of that polynomial.

Important terms used in this statement include the following:

(1) A rational number is any number that has a finite number of digits to the right of the decimal point or any number that has a finite number of digits to the right of the decimal point that repeat in some pattern. Integers are rational numbers.

The fraction 9/11 is a rational number since its decimal representation is 0.818181...; that is, it has an infinite number of digits to the right of the decimal, but those digits repeat.

(2) An irrational number is any number that has an infinite number of digits to the right of the decimal and those digits do not repeat. Examples of irrational numbers include pi and the square roots of prime numbers. Adding, subtracting, multiplying, or dividing an irrational number by a rational number always gives an irrational number. The division by a rational number excludes division by zero.

(3) Let y = a plus the square root of b, where the square root of b is an irrational number. The conjugate of y is a minus the square root of b.

Examples

Using the irrational root theorem, we can answer the question posed at the start of this lesson, which is:

A polynomial with integer coefficients has the following roots:


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Can you find at least two additional roots?

Since the unknown polynomial has integer coefficients, then those coefficients are rational. Since six is not a perfect square, then the square root of 6 is an irrational number. Note that the square root of two is irrational, and two multiplied by an irrational number is also irrational. Furthermore, a rational number plus an irrational number is always irrational.

Thus, two additional roots for the polynomial in question would be:


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