Polynomials

Builds on

Required prerequisites, all the way down: 7

You will learn to

• Add, multiply and factorise polynomials, including special products such as a² − b² = (a − b)(a + b).
• Divide polynomials and use the remainder theorem: the remainder of p(x) ÷ (x − a) is p(a).
• Use the factor theorem to relate roots to linear factors, and bound the number of roots by the degree.

Definition

• A polynomial in x is an expression aₙxⁿ + ⋯ + a₁x + a₀; if aₙ ≠ 0 its degree is n.
• a is a root of p if p(a) = 0.

Theorem

• Division algorithm: for polynomials p and d ≠ 0 with rational (or real) coefficients there are unique q and r with p = qd + r and r = 0 or deg r < deg d.
• Factor theorem: x − a divides p(x) if and only if p(a) = 0.
• A non-zero polynomial of degree n with real or complex coefficients has at most n roots.

Common misconceptions

• Cancelling terms across a fraction: (x² + 1)/x is not x + 1.
• Thinking every polynomial factorises over the integers.
• Believing a polynomial of degree n must have exactly n real roots.

Leads to

Optional discoveries

• Fields and Galois theory Is there a formula with radicals for the roots of every polynomial? For degree five and above there is not (Abel–Ruffini), and Galois theory explains why.
• Numerical methods Most roots cannot be written exactly, but they can be approximated to any accuracy, for instance by Newton's method.

Further reading

• I. M. Gelfand and A. Shen, Algebra (1993)