Common Monomial Factor Calculator

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GCF Monomial

6xy

Factored Form

6xy(2x+3y^2-4x^2y)

Steps

Coefficients: 12, 18, -24
GCD of coefficients = 6
Variable part: min exp of x is 1; min exp of y is 1
GCF = 6xy
Final factorization = 6xy(2x+3y^2-4x^2y)

Term Breakdown

#	Term	Coefficient	Variables	Divided by GCF
1	12x^2y	12	x^2 · y^1	2x
2	18xy^3	18	x^1 · y^3	3y^2
3	-24x^3y^2	-24	x^3 · y^2	-4x^2y

Formula Used

Let each term be \( a_i \prod v_j^{e_{ij}} \). The monomial GCF equals \( \gcd(|a_i|) \cdot \prod v_j^{\min_i e_{ij}} \). Divide each term by the GCF to obtain the factorized expression.

\( \gcd \) over coefficients (integers; decimals are scaled).
For variables, take the minimum exponent across all terms.
Construct \( \text{GCF} \cdot (\text{remaining sum}) \).

How to Use

Enter an expression using + or − between terms.
Use single-letter variables with optional exponents, e.g., x^3y^2.
Press Factor Now to compute the monomial GCF and factorization.
Review steps and term breakdown for transparency.
Download a CSV or PDF report for documentation or sharing.

What is the greatest common monomial factor?

The greatest common monomial factor (GCMF) of several terms is the largest monomial that divides each term.

Take the greatest common divisor of the coefficients.
For every variable present, choose the minimum exponent across all terms.
Multiply these pieces: \( \mathrm{GCMF} = \gcd(|a_i|)\prod v^{\min e_{ij}} \).

Example. For 18x^3y^2, 24x^2y^4, 6xy:

Coefficient gcd = 6.
Min exponents: \(x^1\), \(y^1\).
So GCMF = 6xy. Each term divided by 6xy gives the factorization.

How to find greatest common monomial factor?

Rewrite each term as a coefficient times variable powers.
Compute the gcd of absolute coefficients (scale decimals if needed).
For each variable in all terms, take the minimum exponent.
Multiply the gcd and those variable powers to get the GCMF.
Divide the whole expression by the GCMF to verify the result.

Quick example. For 14x^4y^3 + 21x^2y^5 - 7x^3y^2:

gcd(14, 21, 7) = 7.
Minimum exponents: \(x^2\) from 4,2,3 and \(y^2\) from 3,5,2.
GCMF = 7x^2y^2.
Factored: 7x^2y^2(2x^2y + 3y^3 - x).

Common mistakes and edge cases

Use +/- between terms; commas or semicolons won’t split terms.
Variables are single letters; input like xy means \(x^1y^1\).
Exponents must be integers; negative exponents are allowed, but only positive common exponents count toward the GCMF.
Decimal coefficients are auto-scaled to integers before computing \(\gcd\).
If all coefficients are negative, the calculator returns a negative GCMF for cleaner parentheses.
Zero terms are ignored; empty or unparsable terms trigger a helpful error.

Practice problems (with answers)

Expression	GCF	Factored
28x^4y^3 + 42x^2y^5 - 14x^3y	14x^2y	14x^2y(2x^2y^2 + 3y^4 - x)
18a^3b^2 - 30a^2b^3 + 12ab	6ab	6ab(3a^2b - 5ab^2 + 2)
9m^2n^4 + 6mn^3 + 3m^3n^2	3mn^2	3mn^2(3mn^2 + 2n + m^2)

Example Data

Expression	GCF	Factored
12x^2y + 18xy^3 - 24x^3y^2	6xy	6xy(2xy + 3y^2 - 4x^2y)
15a^2b^3 + 20ab^2 - 5a^3b	5ab^2	5ab^2(3ab + 4a - a^2)
-8x^4y + -12x^3y^2	-4x^3y	-4x^3y(2x - 3y)