Factoring Out a Monomial from a Polynomial- Step-by-Step
What Factoring Out a Monomial Actually Means
Factoring out a monomial is the process of finding what common factor divides evenly into every term of a polynomial. You pull that common piece out front, and what's left goes inside parentheses.
That's it. No magic, no complex theory. Just division in reverse.
If you have 6x³ + 9x², both terms share a factor of 3x². Pull it out and you get 3x²(2x + 3). The original terms are still there—you just rearranged the multiplication.
Monomial vs Polynomial: The Difference
A monomial is a single term. It can be a number, a variable, or a product of numbers and variables.
Examples: 5, x, 7y², 12abc
A polynomial is the sum or difference of two or more monomials.
Examples: 4x + 7, x² - 5x + 6, 2y³ + 8y² - 3y
When you factor out a monomial, you're finding the largest monomial that divides into every term of your polynomial.
The Step-by-Step Process
Step 1: Identify the Common Factor in Each Term
Look at each term separately. Find what number divides evenly into every coefficient. Find what variable factors appear in every term.
Take 12x⁴ + 18x³ + 6x²
- Numbers: 12, 18, and 6. The GCF is 6.
- Variables: x⁴, x³, x². The smallest exponent is x².
- Combined monomial factor: 6x²
Step 2: Divide Each Term by the Common Factor
Divide every term by 6x²:
- 12x⁴ ÷ 6x² = 2x²
- 18x³ ÷ 6x² = 3x
- 6x² ÷ 6x² = 1
Step 3: Write the Factored Form
Put the common factor outside parentheses. Put the quotients inside.
6x²(2x² + 3x + 1)
Verify by distributing: 6x² × 2x² = 12x⁴. 6x² × 3x = 18x³. 6x² × 1 = 6x². You get your original polynomial back.
Examples Worked Out
Example 1: Simple Two-Term Polynomial
Factor: 14y³ + 7y²
Coefficients: 14 and 7. GCF is 7.
Variables: y³ and y². Smallest exponent is y².
Common factor: 7y²
After dividing: 14y³ ÷ 7y² = 2y, 7y² ÷ 7y² = 1
Answer: 7y²(2y + 1)
Example 2: Three-Term Polynomial with Negative Term
Factor: 8a³ - 12a² + 4a
Coefficients: 8, 12, 4. GCF is 4.
Variables: a³, a², a. Smallest exponent is a.
Common factor: 4a
After dividing: 8a³ ÷ 4a = 2a², -12a² ÷ 4a = -3a, 4a ÷ 4a = 1
Answer: 4a(2a² - 3a + 1)
Example 3: No Common Variable
Factor: 15 + 25x
Coefficients: 15 and 25. GCF is 5.
Variables: no variable appears in both terms. You only factor out the number.
Answer: 5(3 + 5x)
Note: You can rewrite this as 5(5x + 3) too. Both are correct, just different order inside the parentheses.
Common Mistakes to Avoid
- Missing a factor: Always check that you've found the greatest common factor. If 4 works but 8 also works, use 8.
- Dropping the term inside: Every term must be accounted for after factoring. If you have three terms originally, you'll have three terms inside the parentheses.
- Forgetting the 1: When a term divides evenly into the common factor, the result is 1, not 0. The 1 stays visible in the parentheses.
- Wrong sign handling: When factoring out a negative, flip all signs inside the parentheses. Factor -6x² + 9x as -3x(2x - 3).
Factoring Out a Monomial vs Other Factoring Methods
| Method | When to Use | Example |
|---|---|---|
| Factor out monomial | Terms share a common factor | 6x² + 9x = 3x(2x + 3) |
| Factor trinomials | ax² + bx + c pattern | x² + 5x + 6 = (x + 2)(x + 3) |
| Difference of squares | a² - b² pattern | x² - 9 = (x + 3)(x - 3) |
| Grouping | Four-term polynomials | ax + ay + bx + by = (a + b)(x + y) |
Factoring out a monomial is usually the first step before trying other methods. Always check for a common factor before moving to more complex techniques.
Getting Started: Your First Problems
Try these problems. Work through each step before checking answers.
Problem 1: Factor 20m⁴ + 15m²
Problem 2: Factor 9x³ - 6x² + 3x
Problem 3: Factor -8y⁵ + 12y³ - 4y²
Answers
Problem 1: 5m²(4m² + 3)
Problem 2: 3x(3x² - 2x + 1)
Problem 3: -4y²(2y³ - 3y + 1)
Quick Reference Checklist
- Find the GCF of all coefficients
- Find the lowest power of each variable present in every term
- Combine into a single monomial factor
- Divide each original term by this factor
- Write the factor outside, quotients inside parentheses
- Multiply back to verify your answer
That's the full process. Practice with 10 problems and you'll have it down. No need to overcomplicate this—it's just finding what divides evenly and rewriting the multiplication.