Common Factor Calculator (Monomials)

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Category Math & Algebra

Polynomial Expression (e.g., 6x^2 + 9x)

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About

Factorization is a fundamental skill in algebra that simplifies complex expressions and solves quadratic equations. This tool identifies the Greatest Common Factor (GCF) of a polynomial expression. It isolates the highest integer that divides all coefficients and selects the lowest exponent for each variable present in every term. Accuracy in this process is critical for reducing algebraic fractions or finding roots of equations. Errors in the initial factorization step often propagate through the entire problem. This calculator provides a breakdown of how the coefficient GCF and variable components are determined separately.

Formulas

The factorization logic follows two main rules. First, the integer coefficients are analyzed using the Euclidean algorithm to find the greatest divisor.

GCF(a, b) = largest integer dividing both

Second, for variables, the rule of exponents dictates that the GCF contains the variable raised to the minimum power found in the expression.

GCF(x^m, xⁿ) = x^min(m,n)

The final expression is rewritten by dividing each original term by the combined GCF.

Reference Data

Expression	GCF Term	Factored Form	Notes
x² + 5x	x	x(x + 5)	Simple variable extraction
6x² − 9x	3x	3x(2x − 3)	Integer GCF is 3
4x³ + 2x²	2x²	2x²(2x + 1)	Lowest power is squared
12a²b − 8ab²	4ab	4ab(3a − 2b)	Multi-variable logic
5x + 7	1	1(5x + 7)	Prime polynomial

Frequently Asked Questions

If the coefficients share no divisors other than 1 (e.g., 3 and 7), the GCF for the coefficients is 1. The calculator will still look for common variables. If neither exist, the polynomial is considered "prime" or irreducible over integers.

Yes. It is standard convention to factor out the negative sign if the leading term is negative. This keeps the term inside the parentheses positive, which is preferred for further operations like graphing or solving.

The tool parses single-variable polynomials efficiently. While it can detect simple multi-variable structures (like ax + ay), complex multi-variable grouping requires a more advanced multivariate engine.