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Rule

Product of a Conjugate Pair of Binomials

The product of two conjugate binomials is the difference of two squares.

$(a + b) (a - b) = a^{2} - b^{2}$

Proof

This identity can be proved by using the Distributive Property to multiply the binomials.

(a + b) (a - b)

Distribute $(a - b)$

a (a - b) + b (a - b)

Distribute $a$

a^{2} - a b + b (a - b)

Distribute $b$

a^{2} - a b + b a - b^{2}

CommutativePropMult

Commutative Property of Multiplication

a^{2} - a b + a b - b^{2}

Add terms

a^{2} - b^{2}

Therefore, the product of a binomial and its conjugate is the difference of two squares.

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