How to Divide a Polynomial by a Monomial in 2 Simple Steps

What are Polynomials and Monomials?

A polynomial is a mathematical expression made by adding, subtracting, or multiplying numbers and variables, with exponents that are whole numbers. It has a limited number of terms. Each term can have numbers, variables (like x, y), or both, multiplied together.

Example:

3𝑥² + 2𝑥 − 5

This polynomial has three terms: 3𝑥², 2𝑥, and -5.

A monomial is the simplest kind of polynomial — it has only one term, like 3𝑥², 2𝑥, or just a number like -5.

Must-Know Before Dividing a Polynomial by a Monomial

Before we begin, the Exponent Rule for Division is a must-know prerequisite. When dividing terms with the same base:

\[\frac{a^m}{a^n} = a^{m – n}\]

This means we keep the base the same and subtract the exponents. We’ll use this rule as we divide each term of the polynomial by the monomial, step by step.

2 Steps to Divide a Polynomial by a Monomial

Dividing a polynomial by a monomial means taking each term of the polynomial and dividing it by the monomial individually.

Step 1: Split the division for each term

\[\frac{6x^3 + 9x^2 – 12x}{3x} = \frac{6x^3}{3x} + \frac{9x^2}{3x} – \frac{12x}{3x}\]

Step 2: Divide coefficients and apply exponent rules

i. \(\frac{6x^3}{3x} = 2x^{3 – 1} = 2x^2\)

ii. \(\frac{9x^2}{3x} = 3x^{2 – 1} = 3x\)

iii. \(\frac{-12x}{3x} = -4\)

Final Answer:

\[\frac{6x^3 + 9x^2 – 12x}{3x} = \frac{6x^3}{3x} + \frac{9x^2}{3x} – \frac{12x}{3x} = 2x^2 + 3x – 4\]

Example Problems: Dividing a Polynomial by a Monomial

Example 1

Divide

(10𝑘² + 40𝑘) ÷ 5𝑘

Answer:

(10𝑘² + 40𝑘) ÷ 5𝑘 = 10𝑘² ÷ 5𝑘 + 40𝑘 ÷ 5𝑘 = 2𝑘 + 8

Example 2

Divide

\[\frac{-15a^5 + 30a^3 – 45a^2}{-5a^2}\]

Answer:

\[\frac{-15a^5 + 30a^3 – 45a^2}{-5a^2} = \frac{-15a^5}{-5a^2} + \frac{30a^3}{-5a^2} + \frac{-45a^2}{-5a^2}\]

Divide each term:

i. \(\frac{-15a^5}{-5a^2} = 3a^{5 – 2} = 3a^3\)

ii. \(\frac{30a^3}{-5a^2} = -6a^{3 – 2} = -6a\)

iii. \(\frac{-45a^2}{-5a^2} = 9\)

Finally,

3𝑎³ − 6𝑎 + 9

Summary: Keys to Dividing a Polynomial by a Monomial

• To divide a polynomial by a monomial, divide each term separately.
• Apply exponent rules: subtract the exponents when dividing terms with the same base.
• Always simplify the final answer.

Tip: Careful with signs! Double-check positive and negative terms.

Additional Math Topics for Algebra 1 – with Free Worksheets

• How to Multiply Polynomials: From 1 Term to 3 Terms

Want more printable practice?

To access more printable worksheets on other math topics, visit this page.

About Think Academy

Think Academy, a leading K–12 math education provider wholly owned by TAL Education Group, is dedicated to helping students build strong mathematical foundations and critical thinking. Our structured curriculum provides multiple course levels designed to accommodate students with diverse academic goals and proficiency levels, ensuring targeted and effective learning experiences. Supported by advanced teaching methods, expert instructors, and innovative AI technology, Think Academy consistently demonstrates excellence, trustworthiness, and proven expertise in mathematics education.

Want more insights on math education and parenting tips? Subscribe to our newsletter for weekly expert advice and updates on the latest learning tools.