The Arithmetic of Polynomials

In practice

Polynomials are the integers of algebra. They add by collecting like terms, multiply by distribution, and factor back into the pieces multiplication came from — x² + 8x + 15 splitting into (x + 3)(x + 5) is 15 splitting into 3 · 5, one level up. Some refuse to split, the way primes refuse: x² + 9 stands whole over the real numbers. The factoring strategy is a fixed patrol — GCF first, count the terms, match the pattern, multiply back — and the multiply-back check is the unit's conscience: a factoring that fails it is not a factoring. Rational expressions then replay fractions. They cancel by shared factors only, find common ground at the least common denominator, and divide by flipping the divisor — every move a Unit 1 fraction move with polynomials in the seats, and every move checkable by planting a number: (x² − 9)/(x² + 5x + 6) and its simplified form (x − 3)/(x + 2) both answer −2/3 at x = 1, and agreement like that is the receipt. The unit's three error twins are one error wearing three coats: the missing middle squares term by term, the canceled term cuts across a sum, the denominator split divides across one — each wishes an operation distributed where it does not. (6 + 9)/3 splits to 5 either way; 12/(2 + 4) is 2, and no wish makes it 9. Structure decides what distributes. The wish never does.