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Multiplying monomial by binomial

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Multiplying a Monomial by a Binomial

Multiplying a monomial by a binomial means multiplying a single term across both terms of a two-term expression, using the distributive property: a(b + c) = ab + ac. Learn the exponent rules involved, a worked example, and common mistakes to avoid.

What a monomial and a binomial are

A monomial is a single term, like 3x or 7. A binomial is two terms joined by a plus or minus sign, like 2x + 5. Multiplying a monomial by a binomial means multiplying that single term across both terms of the binomial.

The rule that makes this work is the distributive property: a(b + c) = ab + ac. The monomial "distributes" — it multiplies each term inside the parentheses separately, and the two resulting products are added together. When the terms involve exponents, use the exponent rules to combine them (x × x = x²).

Multiplying a monomial by a binomial using the distributive property The monomial 3x is distributed across each term of the binomial 2x plus 5. 3x times 2x equals 6x squared. 3x times 5 equals 15x. Adding the two products gives 6x squared plus 15x. 3x( 2x + 5 ) 3x × 2x = 6x² 3x × 5 = 15x + 3x(2x + 5) = 6x² + 15x
Distributing 3x across (2x + 5): 3x × 2x = 6x², and 3x × 5 = 15x, so 3x(2x + 5) = 6x² + 15x.

Multiply 4x(3x − 2):

  • 4x × 3x = 12x²
  • 4x × (−2) = −8x
  • So 4x(3x − 2) = 12x² − 8x.
  • Forgetting the sign: when the binomial has a minus sign, that sign travels with the second term into the product.
  • Only multiplying the first term: the monomial must be distributed to every term inside the parentheses, not just the first one.
  • Adding exponents incorrectly: x × x = x², not 2x — multiplying like bases adds their exponents.

This is the building block for multiplying a binomial by a binomial, which applies the same distributive idea twice. The simpler case — one term times one term — is covered in multiplying a monomial by a monomial.

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