Factoring
AlgebraFactoring 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. It is the reverse of multiplying.
Example
Factor $x^2 + 5x + 6$. Since $2 \cdot 3 = 6$ and $2 + 3 = 5$, the answer is $(x + 2)(x + 3)$. Check: $(x+2)(x+3) = x^2 + 5x + 6$.
Key Insight
Factoring is like "unmultiplying." Just as $12 = 3 \cdot 4$, many polynomials can be broken into factors.
Definition
Factoring a polynomial over a field means expressing it as a product of lower-degree polynomials (or irreducible polynomials). The first step is always to factor out the greatest common factor (GCF). Then look for special patterns: difference of squares, perfect square trinomial, or factor by grouping.
Example
Factor $3x^3 - 12x$: GCF $= 3x$, giving $3x(x^2 - 4)$. Recognize $x^2 - 4$ as difference of squares: $3x(x+2)(x-2)$.
Key Insight
The factored form of a polynomial directly reveals its zeros (roots). If $f(x) = (x-2)(x+3)$, the zeros are $x = 2$ and $x = -3$. Factoring is the foundation for solving polynomial equations.
Definition
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, each $p_i$ is a monic irreducible polynomial, and $e_i \ge 1$. Over $\mathbb{Q}$, this is unique by the unique factorization property of $\mathbb{Q}[x]$. Over $\mathbb{C}$, every non-constant polynomial factors completely into linear factors (Fundamental Theorem of Algebra). Over $\mathbb{R}$, irreducible polynomials have degree $1$ or $2$.
Example
$x^4 - 1$ factors over $\mathbb{R}$ as $(x^2-1)(x^2+1) = (x-1)(x+1)(x^2+1)$. Over $\mathbb{C}$: $(x-1)(x+1)(x-i)(x+i)$.
Key Insight
Unique factorization in polynomial rings mirrors prime factorization in integers. Galois theory answers which polynomials can be factored by radicals, connecting factoring to group theory and the solvability of groups.