1Cademy - Multiplying $$(3pq + 5)(6pq - 11)$$ Using the FOIL Method

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The FOIL Method for Multiplying Binomials

Example

Multiplying $(3pq + 5)(6pq - 11)$ Using the FOIL Method

Multiply $(3pq + 5)(6pq - 11)$ using the FOIL method — both binomials contain two-variable monomial terms ( $pq$ ), so each FOIL multiplication involves multiplying monomials with multiple variables.

Step 1 — First: Multiply the first terms of each binomial: $3pq \cdot 6pq = 18p^2q^2$ (multiply the coefficients $3 \cdot 6 = 18$ and apply the Product Property to each variable: $p \cdot p = p^2$ and $q \cdot q = q^2$ ).

Step 2 — Outer: Multiply the outermost terms: $3pq \cdot (-11) = -33pq$ .

Step 3 — Inner: Multiply the innermost terms: $5 \cdot 6pq = 30pq$ .

Step 4 — Last: Multiply the last terms of each binomial: $5 \cdot (-11) = -55$ .

Writing all four products in order gives:

$18p^2q^2 - 33pq + 30pq - 55$

Step 5 — Combine like terms: The Outer and Inner products $-33pq$ and $30pq$ are like terms because both contain the same variable structure $pq$ . Combine their coefficients: $-33 + 30 = -3$ :

$18p^2q^2 - 3pq - 55$

The result is $18p^2q^2 - 3pq - 55$ . When both binomials contain two-variable terms like $pq$ , the First product involves applying the Product Property for Exponents to each variable independently — $p \cdot p = p^2$ and $q \cdot q = q^2$ . Despite the added complexity of two variables, the Outer and Inner products remain like terms (both are multiples of $pq$ ), so the result simplifies to a trinomial.

Updated 2026-04-29

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