Factoring
Factoring is the process of rewriting a polynomial as a product of simpler expressions (its factors), reversing the process of multiplication.
Definition
Factoring means breaking a polynomial into smaller pieces that multiply together to give the original, the reverse of multiplying. The first step is always to pull out the greatest common factor (GCF), then look for special patterns, difference of squares, perfect square trinomial, or factor by grouping. Formally, factoring a polynomial $f$ in $F[x]$ means writing $f = c \cdot p_1^{e_1} \cdot \ldots \cdot p_k^{e_k}$, where $c$ is a constant and each $p_i$ is a monic irreducible polynomial; over $\mathbb{Q}$ this factorization is unique, over $\mathbb{C}$ every non-constant polynomial factors completely into linear factors (the Fundamental Theorem of Algebra), and over $\mathbb{R}$ irreducible factors have degree $1$ or $2$.
Example
Factoring $x^2 + 5x + 6$: since $2 \cdot 3 = 6$ and $2 + 3 = 5$, the answer is $(x + 2)(x + 3)$. Factoring $3x^3 - 12x$: GCF is $3x$, giving $3x(x^2 - 4)$, and $x^2 - 4$ is a difference of squares, so the full factorization is $3x(x+2)(x-2)$. $x^4 - 1$ factors over $\mathbb{R}$ as $(x-1)(x+1)(x^2+1)$, and over $\mathbb{C}$ as $(x-1)(x+1)(x-i)(x+i)$.
Key Insight
Factoring is like "unmultiplying": just as $12 = 3 \cdot 4$, many polynomials break into factors, and the factored form directly reveals the polynomial's zeros, if $f(x) = (x-2)(x+3)$, its zeros are $x = 2$ and $x = -3$, making factoring the foundation for solving polynomial equations. Unique factorization in polynomial rings mirrors prime factorization in integers, and Galois theory answers which polynomials can be factored by radicals, connecting factoring to group theory and the solvability of groups.