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Multiplying Polynomial by Monomial

To multiply a polynomial by a monomial, distribute the monomial to every term: multiply the coefficients and add the exponents of matching variables. Learn the step-by-step process with a worked example, how to handle negative terms, and how this skill extends to multiplying two polynomials together.

Multiplying a polynomial by a monomial

To multiply a polynomial by a monomial, use the distributive property: multiply the monomial by every term inside the polynomial, then add the results together. This is the same distributive property used for multiplying monomial by binomial, just extended to more terms.

Distributing 3x across (2x² + 4x − 5): 3x×2x² + 3x×4x + 3x×(−5) = 6x³ + 12x² − 15x.

The steps

Multiply the monomial by each term one at a time: multiply the coefficients together, and add the exponents of matching variables. In the example, 3x × 2x² multiplies 3×2 = 6 and adds exponents x¹ × x² = x³, giving 6x³. Repeat for every term, then combine the results — there is usually nothing left to combine further since each product has a different power of x.

Watch the signs

Keep track of negative terms carefully: multiplying a monomial by a negative term inside the polynomial gives a negative result, as with 3x × (−5) = −15x above.

Why this matters

This same distribution skill scales up directly to multiplying polynomial by polynomial and to applications of polynomials in area and volume problems, where expressions need to be expanded before they can be simplified or solved.

Related lessons

Algebra