Polynomial Factorization Calculator
This powerful polynomial factorization calculator can factor polynomial expressions of any degree into their simplest factored forms. Supports quadratic, cubic, and higher-degree polynomials with detailed step-by-step solutions and mathematical formula visualization.
Support Various Polynomials
Handle quadratic, cubic, and higher-degree polynomials, automatically recognizing special forms like perfect squares and difference of squares
LaTeX Formula Display
Beautiful mathematical formula presentation using LaTeX format for easy reading and understanding
Detailed Factorization Steps
Show complete factorization process including common factor extraction, grouping method, and special formula applications
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Enter your polynomial expression and get instant factorization results
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What is Polynomial Factorization?
Understanding the fundamentals of algebraic factorization
Polynomial Factorization is an important operation in algebra, referring to expressing a polynomial as a product of two or more polynomials. Factorization is a fundamental skill in handling algebraic expressions, with significant importance for solving equations, simplifying fractions, and understanding function behavior.
For example, factoring the quadratic polynomial x² - 5x + 6 into (x - 2)(x - 3) is a factorization process. The factored form can intuitively show the roots (zeros) of the polynomial. In this example, x = 2 and x = 3 are the roots of the original polynomial.
Key Concepts
Example: x² - 5x + 6
Verification: (x - 2)(x - 3) = x² - 3x - 2x + 6 = x² - 5x + 6 ✓
Polynomial Factorization Methods
Six powerful approaches to factor polynomials
There are various methods for polynomial factorization, and different strategies can be chosen based on the type and complexity of the polynomial:
Extracting Common Factors
The most basic method is to extract the common factor from all terms. For example: 2x³ + 4x² = 2x²(x + 2), where 2x² is the common factor of both terms.
Using Formula Method
For polynomials of specific forms, standard formulas can be used. For quadratic polynomial ax² + bx + c, use: ax² + bx + c = a(x - r₁)(x - r₂), where r₁ and r₂ are the roots.
Grouping Method
For polynomials with four or more terms, try grouping. Example: x³ + 2x² + 2x + 4 = x²(x + 2) + 2(x + 2) = (x + 2)(x² + 2)
Cross Multiplication
For quadratics ax² + bx + c, find numbers m and n such that m × n = a × c and m + n = b. This helps identify factors efficiently.
Special Formulas
Apply standard formulas: perfect squares a² + 2ab + b² = (a + b)², difference of squares a² - b² = (a + b)(a - b), and cube formulas.
Factor Theorem
If r is a root of polynomial P(x), then (x - r) is a factor of P(x). Use this theorem to construct factorization from known roots.
Polynomial Factorization Examples
Step-by-step solutions to master the concepts
Example 1: Quadratic Polynomial
Factor x² - 7x + 12
Step 1: Find two numbers whose product is 12 (the c value) and sum is -7 (the b value)
Step 2: These numbers are -3 and -4, because (-3)×(-4)=12 and (-3)+(-4)=-7
Step 3: Therefore, x² - 7x + 12 = (x - 3)(x - 4)
Example 2: Polynomial with Common Factor
Factor 3x² - 12
Step 1: Extract the common factor: 3x² - 12 = 3(x² - 4)
Step 2: Use the difference of squares formula to factor x² - 4 = (x + 2)(x - 2)
Step 3: Therefore, 3x² - 12 = 3(x + 2)(x - 2)
Example 3: Higher Degree Polynomial
Factor x³ - 8
Step 1: Use the difference of cubes formula: x³ - 8 = x³ - 2³ = (x - 2)(x² + 2x + 4)
Step 2: Since x² + 2x + 4 cannot be factored further in the real domain, the final result is (x - 2)(x² + 2x + 4)
Frequently Asked Questions
Common questions about polynomial factorization