How to Multiply a Polynomial and a Monomial? (+FREE Worksheet!)
Multiplying a polynomial and a monomial is one of the foundational operations in Algebra 1. The key tool is the distributive property, which tells you to multiply the monomial by every single term inside the polynomial. This page explains the method clearly, shows four fully worked examples, and provides practice problems so you can gain fluency.
Multiply a Polynomial and a Monomial: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Organize by degreeWrite terms from highest power to lowest power.
- Look for structureTry GCF, special products, grouping, or division depending on the expression.
- Check with featuresZeros, multiplicity, and end behavior should agree with your algebra.
Worked examples
Combine like terms
- Group x squared terms.
- Group x terms.
- Combine each group.
Factor a difference of squares
- Recognize a squared term minus a squared term.
- Use a^2 – b^2.
- Write conjugate factors.
Try one before moving on
Multiply a Polynomial and a Monomial: pop-up practice
What Is the Distributive Property?
The distributive property states that for any values a, b, and c:
\(\color{blue}{a(b + c) = \text{ ab } + \text{ ac }}\)
To multiply a monomial by a polynomial, you distribute the monomial to every term in the polynomial. Each multiplication follows the monomial multiplication rules: multiply coefficients and add exponents.
How to Multiply a Polynomial by a Monomial
1. Distribute to Every Term
Multiply the monomial by the first term, then the second term, and so on, one at a time.
- \(\color{blue}{3x(2x^{2} – 4x + 5) = 3x \cdot 2x^{2} + 3x \cdot (-4x) + 3x \cdot 5}\)
2. Simplify Each Product
Multiply coefficients together and add the exponents of matching variables.
- \(\color{blue}{3x \cdot 2x^{2} = 6x^{3}}\)
- \(\color{blue}{3x \cdot (-4x) = -12x^{2}}\)
- \(\color{blue}{3x \cdot 5 = 15x}\)
Result: \(\color{blue}{6x^{3} – 12x^{2} + 15x}\)
3. Combine Any Like Terms
After distributing, check whether any of the resulting terms are like terms and combine them if so. In most monomial-times-polynomial problems the terms will already be unlike, so no further combining is needed.
Step-by-Step Summary
- Write the monomial outside the parentheses.
- Multiply the monomial by every term inside the polynomial.
- For each product: multiply coefficients, add exponents of same-base variables.
- Keep terms with different variables separate.
- Combine like terms if any exist, then write in descending order.
Watch: Multiplying a Monomial by a Polynomial (Video Lesson)
Khan Academy demonstrates the distributive approach with a clear example:
Worked Examples
Example 1: Multiply \(\color{blue}{3x(2x^{2} – 4x + 5)}\).
Distribute: \(\color{blue}{3x \cdot 2x^{2} + 3x \cdot (-4x) + 3x \cdot 5}\)
Simplify: \(\color{blue}{6x^{3} – 12x^{2} + 15x}\)
Example 2: Multiply \(\color{blue}{-2a^{2}(3a^{2} – a + 4)}\).
Distribute: \(\color{blue}{(-2a^{2})(3a^{2}) + (-2a^{2})(-a) + (-2a^{2})(4)}\)
Simplify: \(\color{blue}{-6a^{4} + 2a^{3} – 8a^{2}}\)
Example 3: Multiply \(\color{blue}{4\text{ xy }(2x^{2}y – 3\text{ xy }^{2} + y)}\).
Distribute each term:
\(\color{blue}{4\text{ xy } \cdot 2x^{2}y = 8x^{3}y^{2}}\)
\(\color{blue}{4\text{ xy } \cdot (-3\text{ xy }^{2}) = -12x^{2}y^{3}}\)
\(\color{blue}{4\text{ xy } \cdot y = 4\text{ xy }^{2}}\)
Result: \(\color{blue}{8x^{3}y^{2} – 12x^{2}y^{3} + 4\text{ xy }^{2}}\)
Example 4: Multiply \(\color{blue}{-4x^{2}(3x^{2} – 2x + 1)}\).
Distribute:
\(\color{blue}{(-4x^{2})(3x^{2}) = -12x^{4}}\)
\(\color{blue}{(-4x^{2})(-2x) = 8x^{3}}\)
\(\color{blue}{(-4x^{2})(1) = -4x^{2}}\)
Result: \(\color{blue}{-12x^{4} + 8x^{3} – 4x^{2}}\)
Area Model Video, Another Approach
The area model is a visual way to organize \(\color{blue}{\text{ monomial } \times \text{ polynomial }}\) multiplication. Khan Academy shows the method here:
Exercises for Multiplying a Polynomial and a Monomial
Find each product.
- \(\color{blue}{5x(x^{2} – 3x + 2)}\)
- \(\color{blue}{-3y^{2}(2y – 1)}\)
- \(\color{blue}{2m(m^{2} + 4m – 6)}\)
- \(\color{blue}{-4x^{2}(3x^{2} – 2x + 1)}\)
- \(\color{blue}{3\text{ ab }(a^{2} – 2\text{ ab } + b^{2})}\)
Answers
- \(\color{blue}{5x^{3} – 15x^{2} + 10x}\)
- \(\color{blue}{-6y^{3} + 3y^{2}}\)
- \(\color{blue}{2m^{3} + 8m^{2} – 12m}\)
- \(\color{blue}{-12x^{4} + 8x^{3} – 4x^{2}}\)
- \(\color{blue}{3a^{3}b – 6a^{2}b^{2} + 3\text{ ab }^{3}}\)
Free Multiplying a Polynomial and a Monomial Worksheet
Ready to practice on your own? Download our free Multiplying a Polynomial and a Monomial worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again, steady practice is the surest way to master Multiplying a Polynomial and a Monomial before a quiz or test.
Download Multiplying Polynomials Worksheet
Frequently Asked Questions
How do you multiply polynomials and monomials?
Use the distributive property: multiply the monomial by every term of the polynomial, then write the result in standard form. Multiply coefficients and add exponents of matching variables. For example, 3x(2x² − 4x + 5) = 6x³ − 12x² + 15x. Count the terms: a three-term polynomial times a monomial should give three terms.
Is a monomial a polynomial?
Yes. A monomial is a polynomial with exactly one term, such as 7, 5x, or −2a³b. Polynomials are named by their number of terms: one term is a monomial, two terms is a binomial, and three terms is a trinomial. An expression with a variable in a denominator, like 3/x, is not a monomial or a polynomial.
What is the difference between a monomial and a polynomial?
A monomial has a single term, while a polynomial can have one or more terms joined by addition or subtraction. So 4x² is a monomial, and 4x² − 9x + 1 is a polynomial (a trinomial). Every monomial is a polynomial, but not every polynomial is a monomial. That is why multiplying them takes distribution across each term.
How do you multiply a monomial by a binomial?
Distribute the monomial to both terms of the binomial. For 8x(6x + 6), multiply 8x × 6x = 48x² and 8x × 6 = 48x, so the answer is 48x² + 48x. Watch the signs with subtraction: −2y(y − 5) = −2y² + 10y, because a negative times a negative is positive.
Is multiplying two polynomials a polynomial?
Yes. Polynomials are closed under multiplication, so multiplying any two polynomials always gives another polynomial. For instance, (x + 1)(x − 1) = x² − 1 and 2x(x² + 3) = 2x³ + 6x. The degree of the product equals the sum of the degrees: here 1 + 2 = 3, so 2x³ + 6x is a cubic.
Can you multiply monomials that are not like terms?
Yes. Only addition and subtraction need like terms; multiplication does not. Multiply the coefficients and keep every variable: 4x × 3y = 12xy and 2x² × 5x = 10x³. That is why 5x(x² + 2y) is fine to expand into 5x³ + 10xy even though x³ and xy can’t be combined afterward.
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