Polynomial Arithmetic and Closure

Think of each term as a labeled pile: an "\(x^2\) pile," an "\(x\) pile," a "constant pile." Adding or subtracting two polynomials just merges matching piles — you never mix an \(x^2\) pile with an \(x\) pile, the same way you wouldn't add 3 apples and 2 oranges and call it "5 apple-oranges." Multiplying is different: every term in the first polynomial has to visit every term in the second, so a small polynomial times a small polynomial can produce a much longer one before it's sorted back into piles.

For addition and subtraction, only coefficients of matching-degree terms combine — \(a x^n + b x^n = (a+b)x^n\) — because different powers of \(x\) can't merge. For multiplication, distribute every term of the first factor over every term of the second and combine like terms afterward; since \(x^m \cdot x^n = x^{m+n}\), multiplying a degree-\(m\) polynomial by a degree-\(n\) polynomial always yields a degree-\((m+n)\) result. None of these three operations can ever produce a negative or fractional exponent, so the polynomials are closed under addition, subtraction, and multiplication — unlike division, where \(p(x) \div q(x)\) need not be a polynomial at all.

To add or subtract: line up like terms (distribute a leading minus sign across every term first if you're subtracting), then combine each column's coefficients. To multiply: distribute every term of the first polynomial across every term of the second, then gather the products by degree and combine like terms — exactly what the board above does, one column at a time.