Free factoring calculator. Factor quadratic, cubic, and quartic polynomials step by step. Also find prime factorization and factor pairs of any number.
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Factoring Calculator, Math, Free factoring calculator. Factor quadratic, cubic, and quartic polynomials step by step. Also find prime factorization and factor pairs of any number., factor polynomial, prime factorization, find factors, factor expression, calc, compute
Factoring Calculator
Free factoring calculator. Factor quadratic, cubic, and quartic polynomials step by step. Also find prime factorization and factor pairs of any number.
factor polynomial, prime factorization, find factors, factor expression
Math global
Factoring Calculator, Math, Free factoring calculator. Factor quadratic, cubic, and quartic polynomials step by step. Also find prime factorization and factor pairs of any number., factor polynomial, prime factorization, find factors, factor expression, calc, compute
Factoring Calculator
Free factoring calculator. Factor quadratic, cubic, and quartic polynomials step by step. Also find prime factorization and factor pairs of any number.
Factor
a
x² +
b
x +
c
x² + 5x + 6
Trinomial Factoring
x² + 5x + 6 =
(x + 3)(x + 2)
Degree 2
Factorable
Factoring Details
Original, factored form & method breakdown
Original Expression
x² + 5x + 6
Factored Form
(x + 3)(x + 2)
x² + 5x + 6(x + 3)(x + 2)
Method
Trinomial Factoring
Degree
2
GCF
None
Step-by-Step Solution
How the expression was factored
1
Original expression
x² + 5x + 6
Identify the quadratic expression to factor.
2
AC Method
a × c = 1 × 6 = 6
Find two integers m, n where m × n = 6 and m + n = 5.
3
Found integers
m = 3, n = 2
3 × 2 = 6 ✔, 3 + 2 = 5 ✔
4
Write in factored form
(x + 3)(x + 2)
(x + 3)(x + 2) = x² + 5x + 6 ✔
What Is Factoring?
Breaking expressions into simpler multiplied parts
Factoring is the process of rewriting a mathematical expression as a product of simpler expressions. For example, the quadratic x² + 5x + 6 can be written as (x + 2)(x + 3). This is the reverse of expanding or multiplying out brackets.
For numbers, factoring means finding all integers that divide evenly into a given number. For instance, the factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.
Key Idea
If A = B × C, then B and C are factors of A.
Common Factoring Methods
Six essential techniques for factoring polynomials
1. Greatest Common Factor (GCF)
Always start by factoring out the largest number and variable power common to all terms.
6x³ + 12x² + 18x = 6x(x² + 2x + 3)
2. Difference of Squares
When you have a² − b², it factors as (a + b)(a − b). Both terms must be perfect squares.
x² − 9 = (x + 3)(x − 3)
3. Perfect Square Trinomial
a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)². Check if the middle term equals 2 × √(first) × √(last).
x² + 6x + 9 = (x + 3)²
4. Trinomial Factoring (AC Method)
For ax² + bx + c, find two numbers m and n where m × n = ac and m + n = b. This works for both a = 1 and a ≠ 1.
2x² + 7x + 3 = (2x + 1)(x + 3)
5. Sum & Difference of Cubes
a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). Remember the SOAP mnemonic: Same, Opposite, Always Positive.
x³ − 8 = (x − 2)(x² + 2x + 4)
6. Rational Root Theorem
For higher-degree polynomials, test possible rational roots ±(factors of constant) ÷ (factors of leading coefficient). When a root is found, use synthetic division to reduce the degree.
x³ − 6x² + 11x − 6 = (x − 1)(x − 2)(x − 3)
Worked Examples
Practice recognizing factoring patterns
Expression
Factored Form
Method
x² + 7x + 12
(x + 3)(x + 4)
Trinomial
x² − 16
(x + 4)(x − 4)
Diff. of Squares
4x² + 12x + 9
(2x + 3)²
Perfect Square
3x² + 10x − 8
(3x − 2)(x + 4)
AC Method
x³ + 27
(x + 3)(x² − 3x + 9)
Sum of Cubes
x³ − 4x² − 7x + 10
(x − 1)(x + 2)(x − 5)
Rational Root
Common Mistakes to Avoid
Pitfalls students frequently encounter
Forgetting to extract GCF first
Always check for a common factor before trying other methods. 6x² + 12x + 6 has a GCF of 6.
Sum of squares is NOT factorable
x² + 9 ≠ (x + 3)(x − 3). Only the DIFFERENCE of squares factors. x² + 9 is prime over the reals.
Sign errors in AC method
When ac is negative, one of m, n is positive and the other is negative. Double-check signs.
Not factoring completely
After one factoring step, check if any factor can be factored further. x⁴ − 16 = (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2).
Frequently Asked Questions
Common questions about factoring expressions and numbers
Factoring means rewriting a mathematical expression as a product of simpler expressions. For example, x² + 5x + 6 factors into (x + 2)(x + 3). This is the reverse of expanding brackets. Factoring helps solve equations, simplify fractions, and identify roots (zeros) of polynomials.
First, extract any Greatest Common Factor (GCF) from all terms. Then check for special patterns: difference of squares (a² − b²) or perfect square trinomial (a² ± 2ab + b²). If neither applies, use the AC method: find two numbers m and n where m × n = a × c and m + n = b. If a = 1, the factored form is (x + m)(x + n). If a ≠ 1, use grouping to find (px + q)(rx + s).
Factoring numbers means finding all integers that divide evenly into a given number (e.g., factors of 12 are 1, 2, 3, 4, 6, 12). Factoring expressions means rewriting a polynomial as a product of lower-degree polynomials (e.g., x² − 9 = (x + 3)(x − 3)). Both involve breaking something into its multiplicative components.
Prime factorization expresses a number as a product of prime numbers. Every integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic). For example, 48 = 2⁴ × 3. To find it, repeatedly divide by the smallest prime that divides evenly until you reach 1.
The AC method works for any quadratic ax² + bx + c. Multiply a × c to get the product. Find two integers m and n where m × n = ac and m + n = b. If a = 1, write directly as (x + m)(x + n). If a ≠ 1, rewrite bx as mx + nx, then factor by grouping. The method always works when integer factors exist.
Not every polynomial can be factored over the integers. For example, x² + x + 1 has no integer factors. However, every polynomial with real coefficients can be factored into linear and irreducible quadratic factors over the reals. The discriminant b² − 4ac tells you if a quadratic is factorable: if it is a non-negative perfect square, the quadratic factors over the integers.
The difference of squares formula states a² − b² = (a + b)(a − b). It only works for subtraction, not addition. For example, x² − 25 = (x + 5)(x − 5), but x² + 25 cannot be factored over the reals. Always check if both terms are perfect squares before applying this formula.
For cubics (ax³ + bx² + cx + d), first extract any GCF and check for a common factor of x (if d = 0). Then check for sum/difference of cubes patterns (a³ ± b³). If none apply, use the Rational Root Theorem to test possible roots (±factors of d ÷ factors of a). When you find a root r, divide by (x − r) using synthetic division to get a quadratic, then factor that quadratic.
The formulas are: a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²). Remember the SOAP mnemonic: Same sign, Opposite sign, Always Positive. For example, x³ − 8 = (x − 2)(x² + 2x + 4) where a = x and b = 2.
Follow this checklist: (1) Always extract GCF first. (2) Count the terms: 2 terms → check difference of squares or cubes; 3 terms → check perfect square trinomial, then try AC method; 4+ terms → try grouping or rational root theorem. (3) After each factoring step, check if any factor can be factored further. Practice helps you recognize patterns faster.
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