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

Moving from basic algebra to polynomial operations can feel like a big step, but it becomes easier once students understand the basics. In Algebra 1, students learn how to work with polynomials and monomials because they appear in many expressions and equations.

Dividing a polynomial by a monomial is an algebra skill that is introduced in Pre-Algebra as a preview and developed further in Algebra 1.

This guide explains polynomials and monomials in simple terms and shows how to divide a polynomial by a monomial step by step. Using key rules like the exponent rule, students can simplify expressions in a clear and repeatable way.

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.

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More related articles

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

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