Pengi Editor's Note: This article was originally published by Think Academy. We're sharing it here for educational value. Think Academy is a leading K-12 math education provider.
How to Divide a Polynomial by a Monomial in 2 Simple Steps
Dividing a polynomial by a monomial is one of the foundational operations in algebra. It follows directly from the properties of exponents and the distributive property.
Key Concept
When you divide a polynomial by a monomial, you divide each term of the polynomial by the monomial separately.
$$\frac{a + b + c}{d} = \frac{a}{d} + \frac{b}{d} + \frac{c}{d}$$
The 2-Step Process
Step 1: Divide each term of the polynomial by the monomial.
Step 2: Simplify each resulting term using the rules of exponents.
Rules of Exponents (Review)
- $$x^m \div x^n = x^{m-n}$$
- $$\frac{x^5}{x^2} = x^3$$
- $$\frac{x^2}{x^2} = x^0 = 1$$
- $$\frac{x^1}{x^3} = x^{-2} = \frac{1}{x^2}$$
Example 1
$$\frac{6x^3 + 4x^2 - 2x}{2x}$$
Step 1: Divide each term by 2x:
$$\frac{6x^3}{2x} + \frac{4x^2}{2x} - \frac{2x}{2x}$$
Step 2: Simplify:
$$\frac{6}{2} \cdot x^{3-1} + \frac{4}{2} \cdot x^{2-1} - \frac{2}{2} \cdot x^{1-1}$$
$$= 3x^2 + 2x - 1$$
Answer: 3x² + 2x − 1
Example 2
$$\frac{15x^4 - 10x^3 + 5x^2}{5x^2}$$
Step 1: Divide each term by 5x²:
$$\frac{15x^4}{5x^2} - \frac{10x^3}{5x^2} + \frac{5x^2}{5x^2}$$
Step 2: Simplify:
$$= 3x^2 - 2x + 1$$
Answer: 3x² − 2x + 1
Example 3: With Multiple Variables
$$\frac{12x^3y^2 + 8x^2y - 4xy}{4xy}$$
Step 1: Divide each term by 4xy:
$$\frac{12x^3y^2}{4xy} + \frac{8x^2y}{4xy} - \frac{4xy}{4xy}$$
Step 2: Simplify each term:
$$= \frac{12}{4} \cdot x^{3-1} \cdot y^{2-1} + \frac{8}{4} \cdot x^{2-1} \cdot y^{1-1} - 1$$
$$= 3x^2y + 2x - 1$$
Answer: 3x²y + 2x − 1
Example 4: With Negative Exponent
$$\frac{9x^3 - 6x}{3x^2}$$
$$= \frac{9x^3}{3x^2} - \frac{6x}{3x^2}$$
$$= 3x - \frac{2}{x}$$
Answer: 3x − 2/x
(Note: 6x ÷ 3x² = 2x⁻¹ = 2/x)
Common Mistakes to Avoid
- Dividing only the first term: Every term in the polynomial must be divided by the monomial.
- Errors with exponents: Remember to subtract exponents when dividing: x⁵ ÷ x² = x³, not x⁷.
- Forgetting to simplify coefficients: 12/4 = 3, not left as 12/4.
- Sign errors: Be careful when terms are negative — negative ÷ positive = negative.
Practice Problems
- (8x⁴ + 6x³ − 2x²) ÷ 2x²
- (15x³ − 10x² + 5x) ÷ 5x
- (12x²y − 8xy² + 4xy) ÷ 4xy
- (9x³ − 6x² + 3x) ÷ 3x
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