Calculator
Formula Used
The calculator uses a monomial greatest common factor model for algebraic terms.
- Coefficient part: GCF of absolute coefficients using the Euclidean algorithm.
- Variable part: keep variables found in every non-zero term.
- Power rule: for each shared variable, use the smallest exponent.
- Final form: expression = GCF × remaining factor.
For terms 12x^3y^2, 18x^2y, and 30x^2y^3,
the coefficient GCF is 6. The lowest power of x is 2.
The lowest power of y is 1. So the GCF is 6x^2y.
How to Use This Calculator
- Type one expression into the input box.
- Use a new line or semicolon for another expression set.
- Use caret notation for powers, such as
x^3. - Select negative factoring or case-sensitive variables if needed.
- Press Find GCF.
- Review the GCF, factored form, steps, table, and graph.
- Download the CSV or PDF report for later use.
Example Data Table
| Expression | Expected GCF | Reason |
|---|---|---|
| 12x^3y^2 + 18x^2y - 30x^2y^3 | 6x^2y | 6 is common, x^2 is lowest, y is lowest. |
| 15a^3b + 25a^2b^4 - 10a^2b | 5a^2b | All terms share 5, a^2, and b. |
| -8m^4n^2 + 12m^3n - 20m^3n^3 | 4m^3n | Use negative option to factor out -4m^3n. |
| 7p^2q + 14pq^2 + 21pq | 7pq | Each term contains 7, p, and q. |
Understanding GCF of Expressions
A greatest common factor is the largest factor shared by every term. In expressions, it usually has two parts. One part comes from the numbers. The other part comes from the variables. Finding it helps you simplify work before solving, graphing, or comparing algebraic forms.
Why It Matters
Factoring out the GCF makes long expressions easier to read.
It also reveals structure.
For example, 12x^3y^2 + 18x^2y has a shared base.
When 6x^2y is removed, the expression becomes 6x^2y(2xy + 3).
The shorter factor is easier to inspect.
How the Number Part Works
Start with the coefficients. Ignore signs at first. Find the greatest number that divides each coefficient without a remainder. This tool uses the Euclidean algorithm because it is fast and reliable. If the first term is negative, you may choose to factor out a negative GCF.
How Variables Are Compared
Next, compare every variable.
A variable must appear in every non-zero term to enter the GCF.
Then use the smallest exponent found.
In x^5, x^3, and x^4, the common variable part is x^3.
The lowest power controls the factor.
Best Input Practice
Use expanded terms for best results.
Write multiplication by placing the coefficient before variables.
Use ^ for exponents.
Avoid grouped expressions unless they are already expanded.
Review the parsed table when checking complex work.
The graph helps compare coefficient sizes and shared powers quickly.
The export buttons are useful for assignments, lessons, or answer keys.
FAQs
What is the GCF of expressions?
It is the greatest factor shared by every term in an algebraic expression. It can include a number, variables, and the lowest shared powers.
Can this calculator handle powers?
Yes. Enter powers with caret notation, such as x^2 or a^5. The calculator compares powers and keeps the smallest shared exponent.
Does the calculator factor negative expressions?
Yes. Select the negative factor option when the leading term is negative. The GCF will use a negative sign when appropriate.
Can I enter more than one expression?
Yes. Use separate lines or semicolons. The tool finds a shared GCF across all entered non-zero terms and shows each factored line.
Why does a variable disappear from the GCF?
A variable must appear in every non-zero term. If one term does not contain it, that variable cannot be part of the common factor.
Are decimal coefficients supported?
This version is designed for integer coefficients. For decimals, multiply the expression by a power of ten first, then factor the converted expression.
What does the graph show?
The bar chart compares absolute term coefficients. The second chart shows shared variable powers used in the final GCF.
Can I save my result?
Yes. Use the CSV button for spreadsheet data. Use the PDF button for a clean report with the final GCF and steps.