REFERENCE SHEET

Polynomials & roots

Factorization translates an equation into structural information about its roots.

p(x) = (x − a)q(x) + p(a)

Definitions

Polynomial over a field

A finite sum a₀+a₁x+⋯+aₙxⁿ with coefficients in a field F. A nonzero polynomial has degree equal to its highest nonzero exponent.

Root and multiplicity

a is a root if p(a)=0; it has multiplicity m if (x−a)ᵐ divides p but (x−a)ᵐ⁺¹ does not.

Irreducible polynomial

A nonconstant polynomial that cannot be factored into two positive-degree polynomials over the specified field.

Key theorems

Division algorithm

For p,d ∈ F[x] with d≠0, there are unique q,r with p=qd+r and either r=0 or deg r<deg d.

Root bound

A nonzero degree-n polynomial over a field has at most n distinct roots.

Fundamental theorem of algebra

Every nonconstant complex polynomial has a complex root, hence factors into linear factors over ℂ.

Notable applications

Eigenvalues

The roots of det(λI−A) are precisely the eigenvalues of a square matrix.

Interpolation

Distinct sample locations determine a unique polynomial of degree at most n−1 through n prescribed values.

Work it out

Try before revealing

01 / Compute

Factor x³−3x²+3x−1 and give its roots with multiplicities.

Hint

Compare with a binomial cube.

Solution

It is (x−1)³. The only root is 1, of multiplicity 3.

02 / Prove

Why must two degree-at-most-2 polynomials agreeing at three distinct points be identical?