Irrational Conjugate Theorem: Definition & Example

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Irrational numbers lend themselves well to the study of polynomials. This lesson will explain how and why this is by defining the irrational conjugates theorem and showing how we can use it to determine various characteristics of a polynomial.

Irrational Conjugates

Get ready for an extremely simple mathematical definition! Bet you didn't even think such a thing existed, but as we're about to see, these rare creatures do exist! You see, you may already be familiar with an irrational number. An irrational number is a number that extends on forever past its decimal point without taking on any sort of pattern or repetition, and it cannot be written as a fraction. Think of π = 3.14159265... you could spend an eternity writing out its digits without finding a pattern.

Thankfully, many irrational numbers can be written in the form a + √b, where a and b are real numbers, and b is not a perfect square. For example, 2 + √3 or 4 - √5 are irrational numbers.


Now here comes that simple definition! The irrational conjugate of an irrational number a + √b is a - √b. Simple, huh? Basically, we can find the irrational conjugate of an irrational number a + √b by changing addition to subtraction or vice versa.

For example, the irrational conjugate of 2 + √3 is 2 - √3 and the irrational conjugate of 4 - √5 is 4 + √5. Easy Peasy!

Irrational Conjugates Theorem

How about some more simplicity in mathematics? Irrational conjugates have their very own theorem that is just as simple as their definition. The irrational conjugates theorem states that if a + √b is an irrational root to a polynomial, then its irrational conjugate a - √b is also a root.


For example, if 1 + √5 is an irrational root of the polynomial x2 - 2x - 4, then by the irrational conjugates theorem, 1 - √5 must also be a root. Once again, easy peasy!

This theorem proves itself to be very useful in the application and study of polynomials. Believe it or not, we can actually find a polynomial given a limited number of its zeros using this theorem. Let's see how this works!

Using the Theorem to Write Polynomials

A zero of a polynomial, P, is a value of the variable in the polynomial that makes the statement P = 0 true. Zeroes are the same thing as roots of the polynomial, so we will use these two terms interchangeably. Here's where it gets neat! Each zero, a, of a polynomial corresponds to the linear factor x - a in the complete factorization of the polynomial. Therefore, given the zeros, or roots, of a polynomial, we can write out its factors and multiply them together to find the polynomial.

To illustrate this, suppose we know that a polynomial, P, has exactly two zeros: 8 and -9. This tells us that x - 8 and x + 9 are the linear factors in the complete factorization of the polynomial. That is,

  • P = (x - 8)(x + 9)

We now have the polynomial in factored form, and we can multiply it out to find the polynomial in non-factored form.


We end up with P = x2 + x - 72.

Cool! But what does this have to do with the irrational conjugates theorem? Well, consider another example. Suppose we know that a polynomial P has exactly three zeros, and two of them are given as 1 + √2 and 3. Hmmm…we have two of the zeros, so we know we can find two of the linear factors of the polynomial that correspond to these zeros, but we were told that the polynomial has exactly three zeros. How do we find the third? Ah-ha - the irrational conjugates theorem!

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