How to factor in Algebra means rewriting an expression as a product of simpler expressions without changing its value. Factoring is the reverse of distribution, and it exposes structure that supports simplification, solving equations, and analyzing polynomials.
Factoring as reverse distribution
Distribution expands a product into a sum:
Factoring reverses that move:
A correct factorization multiplies back to the original expression exactly. Expanding the proposed factors is the definitive check.
Common factor extraction
Many expressions contain a numerical factor, a variable factor, or both, shared by every term. Pulling out that common factor shortens the expression and often unlocks additional patterns.
The factored form shows the greatest common factor \(6x^2y\) and leaves a simpler binomial.
Standard identities that appear frequently
Several products expand into recognizable sums and differences. These identities provide high-value targets in factoring.
| Pattern in expanded form | Factored form | Notes |
|---|---|---|
| \(A^2 - B^2\) | \((A - B)\cdot(A + B)\) | Difference of squares |
| \(A^2 + 2\cdot A \cdot B + B^2\) | \((A + B)^2\) | Perfect square trinomial |
| \(A^2 - 2\cdot A \cdot B + B^2\) | \((A - B)^2\) | Perfect square trinomial |
| \(A^3 - B^3\) | \((A - B)\cdot(A^2 + A \cdot B + B^2)\) | Difference of cubes |
| \(A^3 + B^3\) | \((A + B)\cdot(A^2 - A \cdot B + B^2)\) | Sum of cubes |
Example of a difference of squares:
Example of a perfect square trinomial:
Trinomials and quadratic products
Quadratic trinomials often factor as a product of two linear factors. A general product expands as:
Matching coefficients connects \(a \cdot x^2 + b \cdot x + c\) with the conditions \(p \cdot q = a\), \(r \cdot s = c\), and \(p \cdot s + q \cdot r = b\).
Worked example:
The check by expansion:
Four-term expressions and grouping structure
Some polynomials factor by organizing terms into two groups with a shared binomial factor.
Visualization of method selection
Common pitfalls
- Equivalence loss: changing a sign or dropping a factor breaks equality; expansion confirms equivalence immediately.
- Incomplete factorization: a common factor left inside parentheses is a partial result; a fully factored form has no further nontrivial common factor.
- Identity mismatch: \(A^2 + B^2\) does not factor over the real numbers into linear factors, while \(A^2 - B^2\) does.
- Trinomial sign errors: the middle term \(b \cdot x\) equals the sum of the cross-products in \((p \cdot x + r)\cdot(q \cdot x + s)\).
Summary
A reliable factoring framework combines common factor extraction with pattern recognition: products that expand to squares, cubes, and trinomials reappear throughout Algebra. A correct factorization remains algebraically identical to the original expression and expands back without discrepancy.