Factor the Expression Completely Calculator

Factoring Methods:

\[ \text{Common Factor: } ax + ay = a(x + y) \] \[ \text{Difference of Squares: } a^2 - b^2 = (a - b)(a + b) \] \[ \text{Quadratic: } x^2 + bx + c = (x + m)(x + n) \]

Factored Form:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What Is Factoring Completely?

Factoring completely means breaking down an algebraic expression into its simplest multiplicative components. This process reveals the expression's roots and is fundamental in solving equations.

2. How Factoring Works

Common factoring methods include:

\[ \text{Greatest Common Factor (GCF): } 6x^2 + 9x = 3x(2x + 3) \] \[ \text{Difference of Squares: } x^2 - 16 = (x - 4)(x + 4) \] \[ \text{Trinomials: } x^2 + 5x + 6 = (x + 2)(x + 3) \]

Process:

Factor out the GCF if possible
Identify special patterns (difference of squares, perfect square trinomials)
Factor remaining expressions
Continue until all factors are prime

3. Importance of Factoring

Applications: Solving equations, simplifying expressions, finding roots/zeros, analyzing polynomial behavior, and calculus operations.

4. Using the Calculator

Instructions: Enter polynomial expressions in standard form (e.g., "x^2-9" or "2x^2+5x-3"). The calculator will attempt to factor completely.

5. Frequently Asked Questions (FAQ)

Q1: What does "factor completely" mean?
A: It means breaking down an expression until all factors are irreducible (can't be factored further).

Q2: Can all expressions be factored?
A: All polynomials can be factored over complex numbers, but some are irreducible over real numbers (e.g., x²+1).

Q3: What's the difference between factoring and expanding?
A: Factoring writes an expression as a product, while expanding writes products as sums.

Q4: How do I factor expressions with higher degrees?
A: For degrees >2, try factoring by grouping, sum/difference of cubes, or rational root theorem.

Q5: Does order matter in factored form?
A: No, (x+2)(x+3) is equivalent to (x+3)(x+2) by the commutative property.