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.
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
What property do you use to multiply a polynomial by a monomial?
You use the distributive property: multiply the monomial by each term of the polynomial, one at a time. Every single term inside the parentheses gets multiplied.
Do you need to simplify after distributing?
After distributing, collect and combine any like terms that may have resulted. In many \(\color{blue}{\text{ monomial } \times \text{ polynomial }}\) problems, no like terms are produced and the distributed result is already fully simplified.
What is the difference between this and FOIL?
FOIL is a specific mnemonic for multiplying two binomials. Multiplying a monomial by a polynomial is simpler — you only distribute from one term, not two.
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