The Arithmetic of Polynomials

Unit 2 is Unit 1's arithmetic replayed on polynomials: they add, multiply, factor, and form fractions the way integers do, and every error twin guards the same border.

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.