Factor The Following Expression Completely Calculator

This calculator helps you factor polynomial expressions completely. Factoring is a fundamental algebra skill that breaks down complex expressions into simpler, multiplied components. The calculator handles common factoring techniques including:

Factoring out the greatest common factor (GCF)
Factoring quadratics (including perfect square trinomials)
Factoring by grouping
Special factoring formulas (difference of squares, sum/difference of cubes)

How to Use This Calculator

Enter your polynomial expression in the input field below. The calculator will attempt to factor it completely using the most appropriate method. For best results:

Use standard algebraic notation (e.g., x² + 5x + 6)
Include all terms of the expression
Use ^ for exponents (e.g., x^2)
Omit spaces between terms (e.g., x^2+5x+6)

The calculator will display the factored form and show the steps used to achieve it. If the expression cannot be factored further, it will indicate that.

How Factoring Works

Factoring is the process of breaking down a polynomial into a product of simpler polynomials. The fundamental factoring techniques are:

Greatest Common Factor (GCF)

Factor out the largest term that divides all terms of the expression. For example:

Example

Original: 6x² + 9x
GCF: 3x
Factored: 3x(2x + 3)

Factoring Quadratics

For quadratic expressions (degree 2), look for two binomials that multiply to give the original expression. Common methods:

Perfect square trinomials: (a + b)² = a² + 2ab + b²
Difference of squares: a² - b² = (a + b)(a - b)
Factoring by inspection: Find two numbers that multiply to c and add to b

Factoring by Grouping

For expressions with four terms, group terms that have common factors and factor by grouping. For example:

Example

Original: xy + xz + yz + z²
Grouped: (xy + xz) + (yz + z²) = x(y + z) + z(y + z)
Factored: (x + z)(y + z)

Special Factoring Formulas

Recognize and apply these special patterns when they appear:

Difference of squares: a² - b² = (a + b)(a - b) Sum of cubes: a³ + b³ = (a + b)(a² - ab + b²) Difference of cubes: a³ - b³ = (a - b)(a² + ab + b²)

Worked Examples

Example 1: Factoring with GCF

Factor completely: 12x³y - 18xy²

Identify GCF of coefficients (12 and 18): 6
Identify GCF of variables: x²y
Combine: 6x²y
Factor: 6x²y(2x - 3y)

Example 2: Factoring Quadratic

Factor completely: x² - 5x + 6

Find two numbers that multiply to 6 and add to -5: -2 and -3
Write as (x - 2)(x - 3)

Example 3: Factoring by Grouping

Factor completely: 2x² + 5x - 3x - 7.5

Group terms: (2x² + 5x) + (-3x - 7.5)
Factor each group: x(2x + 5) - 1.5(2x + 5)
Factor out common binomial: (x - 1.5)(2x + 5)

Example 4: Special Factoring

Factor completely: 8x³ - 27

Recognize difference of cubes pattern: a³ - b³ = (a - b)(a² + ab + b²)
Identify a = 2x and b = 3
Apply formula: (2x - 3)(4x² + 6x + 9)

Frequently Asked Questions

What is the difference between factoring and expanding?: Factoring breaks down an expression into simpler parts, while expanding combines terms into a single polynomial. For example, (x + 2)(x + 3) factors to x² + 5x + 6, and expanding would go the opposite direction.
When should I use factoring?: Factoring is useful for solving equations, simplifying expressions, and finding roots of polynomials. It's particularly important in calculus and higher mathematics.
What if the calculator can't factor my expression?: The calculator uses standard algebraic techniques. If it can't factor your expression, it might be prime (cannot be factored further) or require more advanced methods beyond basic algebra.
Can this calculator handle fractions or decimals?: Yes, the calculator accepts expressions with fractions and decimals. Just enter them in standard algebraic notation (e.g., 1/2x or 0.5x).
Is there a limit to the degree of polynomials I can factor?: The calculator handles polynomials up to degree 4 (quartics) using standard techniques. More complex polynomials may require advanced methods beyond this calculator's scope.