Factoring¶
Recall (doc/postdoc). Not a first course.
Job: write \(p\in K[x]\) as product of irreducibles; zero-product \(\Rightarrow\) roots.
Core¶
Over field \(K\), factor into units \(\times\) irreducibles (unique up to order/units if UFD).
| Pattern (school \(\mathbb{Z}[x]\)) | Form |
|---|---|
| Common factor | \(c\cdot(\cdots)\) |
| Difference of squares | \(a^2-b^2=(a-b)(a+b)\) |
| Sum/diff of cubes | \(a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)\) |
| Trinomial | seek \(x^2+sx+p=(x-r)(x-t)\) |
| Grouping | \(ac+ad+bc+bd=(a+b)(c+d)\) |
Zero-product (integral domain): \(fg=0\Rightarrow f=0\) or \(g=0\) — hence roots from linear factors.
Irreducibility: \(\deg2,3\) irreducible over \(K\) iff no root in \(K\). Eisenstein, reduction mod \(p\) for \(\mathbb{Z}[x]\).
Used later for¶
- Quadratics: complete factorization over \(\mathbb{C}\)
- Galois theory: irreducible \(\Rightarrow\) minimal polynomials
- Partial fractions: factor denominators
- Eigenvalues: factor \(\chi_A(t)\)
Check¶
Is \(x^2+1\) irreducible over \(\mathbb{R}\)? over \(\mathbb{C}\)?
Over \(\mathbb{R}\): yes (no real root). Over \(\mathbb{C}\): no — \((x-i)(x+i)\).