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WASpecialist MathematicsSyllabus dot point

How does admitting complex roots let every polynomial factor completely, and how do real polynomials behave?

Factorise polynomials over the complex numbers using the conjugate root theorem and the fundamental theorem of algebra

WACE Specialist Unit 3 factorisation over C: the fundamental theorem of algebra, the conjugate root theorem for real polynomials, pairing complex roots into real quadratic factors, and finding all roots from one, with a worked example.

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  1. What this dot point is asking
  2. The fundamental theorem of algebra
  3. The conjugate root theorem
  4. Pairing roots into real quadratics
  5. Strategy: from one root to all factors

What this dot point is asking

SCSA wants you to use these two theorems to find every root of a polynomial and write it as a product of linear factors over C\mathbb{C} or real quadratic and linear factors over R\mathbb{R}.

The fundamental theorem of algebra

This is what makes C\mathbb{C} algebraically closed: there is no polynomial equation that fails to have a solution. Over the reals a quadratic with negative discriminant has no roots, but over C\mathbb{C} it has two.

The conjugate root theorem

The proof uses that conjugation distributes over sums and products, so if P(z)=0P(z) = 0 then P(z)‾=P(zˉ)=0\overline{P(z)} = P(\bar z) = 0 when the coefficients are real. The immediate consequence: real polynomials of odd degree have at least one real root, and non-real roots always pair up.

Pairing roots into real quadratics

A conjugate pair multiplies to a real quadratic:

(zβˆ’(a+ib))(zβˆ’(aβˆ’ib))=z2βˆ’2az+(a2+b2).(z - (a + ib))(z - (a - ib)) = z^2 - 2az + (a^2 + b^2).

So a real polynomial factors over R\mathbb{R} into real linear factors (from real roots) and irreducible real quadratics (from conjugate pairs). This is the bridge between the complex factorisation and the real one.

Strategy: from one root to all factors

If you are given or find one non-real root of a real polynomial, you immediately know its conjugate is a root, giving a real quadratic factor. Divide the polynomial by that quadratic to reduce the degree, then factor what remains.