Example

Multiplying (3+5x)(35x)(3 + 5x)(3 - 5x) Using the Product of Conjugates Pattern

Apply the Product of Conjugates Pattern to multiply (3+5x)(35x)(3 + 5x)(3 - 5x). Although the binomials may appear "backwards" because the constant 33 comes first and the variable term 5x5x comes second, they still form a conjugate pair: both share the same first term 33 and the same last term 5x5x, with one using addition and the other subtraction. Use the formula (a+b)(ab)=a2b2(a + b)(a - b) = a^2 - b^2, where a=3a = 3 and b=5xb = 5x.

Step 1 — Square the first term, 33: 32=93^2 = 9.

Step 2 — Square the last term, 5x5x: (5x)2=25x2(5x)^2 = 25x^2. Both the coefficient and the variable must be squared: 52=255^2 = 25 and x2=x2x^2 = x^2.

Step 3 — Write the difference of squares:

(3+5x)(35x)=925x2(3 + 5x)(3 - 5x) = 9 - 25x^2

The product is 925x29 - 25x^2. This example demonstrates that the Product of Conjugates Pattern applies regardless of the order of terms within the binomials. When the constant appears first, the resulting difference of squares begins with a number rather than a variable term — the constant's square comes first, followed by the subtraction of the variable term's square. Recognizing conjugates even when the terms are arranged differently from the standard ax±bax \pm b form is the key takeaway.

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Updated 2026-04-21

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