1Cademy - Multiplying $$(x - 8)(x + 8)$$ Using the Product of Conjugates Pattern

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Product of Conjugates Pattern

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Multiplying $(x - 8)(x + 8)$ Using the Product of Conjugates Pattern

Apply the Product of Conjugates Pattern to multiply $(x - 8)(x + 8)$ . Begin by recognizing the expression as a product of conjugates: both binomials share the same first term $x$ and the same last term $8$ , with one using subtraction and the other addition. This matches the form $(a - b)(a + b) = a^2 - b^2$ , where $a = x$ and $b = 8$ .

Step 1 — Square the first term, $x$ : $x^2$ .

Step 2 — Square the last term, $8$ : $8^2 = 64$ .

Step 3 — Write the difference of squares:

$(x - 8)(x + 8) = x^2 - 64$

The product is $x^2 - 64$ . This example demonstrates the simplest application of the Product of Conjugates Pattern: the variable term has a coefficient of $1$ and the constant is a plain number. The critical first step is recognizing that the two binomials are conjugates — once identified, squaring the first and last terms and writing their difference produces the answer directly, bypassing the four-step FOIL process entirely.

Updated 2026-04-21

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OpenStax Elementary Algebra

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