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Particularly in the realm of complex numbers and irrational numbers, and more specifically when speaking of the roots of polynomials, a conjugate pair is a pair of numbers whose product is an expression of real integers and/or including variables.

A complex number example:

$3 + 2i \;\;\; 3 - 2i$, a product of 13

An irrational example:

$2 + \sqrt{3} \;\;\; 2 - \sqrt{3}$, a product of 1.

Or: $\sqrt{5} \;\;\; -\sqrt{5}$, a product of -25.

Often times, in solving for the roots of a polynomial, some solutions may be arrived at in conjugate pairs.

If the coefficients of a polynomial are all real, for example, any non-real root will have a conjugate pair.

$x^2 + 4 = 0$, has the conjugate pair roots: $2i$ and $-2i$

If the coefficients of a polynomial are all rational, any irrational root will have a conjugate pair.

$x^2 - 5 = 0$, has the conjugate pair roots: $\sqrt{5}$ and $-\sqrt{5}$