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Recipe

Factor a polynomial completely and state each root's multiplicity

Use when: you have a polynomial and need every root, over \mathbb R or over \mathbb C, along with how many times each one repeats

Goal: the polynomial written as a product of irreducible factors, with each root's multiplicity read straight off

Taught in Polynomials: division, roots, and factorization over ℝ and ℂ

Quick recipe

  1. Test small candidate roots first
  2. Divide out that factor by polynomial long division
  3. Repeat on the quotient until it stops factoring further
  4. Decide whether to stop at \mathbb R or continue to \mathbb C
  5. State the final factorization and list every root's multiplicity
  1. 1

    Try plugging in small integers, 0,\pm1,\pm2,\dots, until one makes the polynomial equal zero. The Rational Root Theorem narrows the search further when needed.

    Example

    For q(x)=x^3-3x^2+4: q(-1)=-1-3+4=0, so x=-1 is a root.

    ⚡ Shortcut

    The Rational Root Theorem restricts candidates to \pm(a divisor of the constant term) over (a divisor of the leading coefficient), instead of testing every integer blindly.

    Why: the Factor Theorem guarantees (x-r) divides the polynomial exactly when r is a root, so one successful test already hands over a whole factor.

  2. 2

    Divide the polynomial by (x-r) to get a quotient of one lower degree, and confirm the remainder is genuinely zero.

    Example

    Dividing x^3-3x^2+4 by (x+1) gives quotient x^2-4x+4 with remainder 0.

    Why: long division peels off the known root exactly, leaving a lower-degree polynomial whose remaining roots are still unknown.

  3. 3

    Apply Steps 1 and 2 again to the quotient. Hitting the same root twice raises its multiplicity by one each time.

    Example

    x^2-4x+4=(x-2)^2: testing x=2 succeeds twice in a row, so its multiplicity is 2.

    • the quotient factors further over the reals — repeat Steps 1 and 2 on it again.
    • the quotient is an irreducible quadratic with a negative discriminant — stop dividing over \mathbb R; it only splits further over \mathbb C.

    Why: multiplicity is nothing more than counting how many times the same linear factor divides out before the quotient stops containing it.

  4. 4

    If an irreducible real quadratic factor remains and complete factorization is required, solve it with the quadratic formula; its two roots always form a conjugate pair.

    Example

    \lambda^2+1 has no real root, but over \mathbb C it factors as (\lambda-i)(\lambda+i).

    Why: the Fundamental Theorem of Algebra guarantees a degree-n polynomial has exactly n roots counted with multiplicity, but only once \mathbb C is available.

  5. 5

    Write the polynomial as a product of irreducible factors, then read each root's multiplicity off as the exponent on its linear factor.

    Example

    q(x)=(x+1)(x-2)^2, so x=-1 has multiplicity 1 and x=2 has multiplicity 2.

    Why: this multiplicity is the exact number that becomes an eigenvalue's algebraic multiplicity once this polynomial is a characteristic polynomial.

Common mistakes

  • Stopping after finding one root instead of dividing out and testing the quotient for repeats.
  • Reporting a repeated root only once when its factor's exponent is greater than one.
  • Treating an irreducible real quadratic as unfactorable instead of splitting it over \mathbb C.
  • A sign slip while testing a candidate root, especially confusing p(-r) with -p(r).