Complex Conjugate Root Theorem
For a complex number , its complex conjugate is . Geometrically, this is the reflection of across the real axis in the Argand diagram.
One of the most important results in polynomial theory is the Complex Conjugate Root Theorem: if a polynomial has real coefficients and (where ) is a root, then its conjugate is also necessarily a root.
Why does this happen? Consider a polynomial with real coefficients. If is a root, then . Taking the conjugate of both sides and using the fact that conjugation distributes over addition and multiplication — and that conjugating a real coefficient leaves it unchanged — gives . So is also a root.
This theorem applies only when all coefficients are real. A polynomial with complex coefficients does not need to have conjugate pairs of roots — for example, has only the root .
A direct consequence: a polynomial with real coefficients can only have an even number of non-real complex roots, since they always appear in conjugate pairs.
A simple verification: the polynomial has roots and , which are conjugates. We can confirm: . ✓
Using Conjugate Roots to Factorise Polynomials
Because complex roots come in conjugate pairs for real-coefficient polynomials, knowing one complex root immediately gives you another — and their product forms a real quadratic factor.
If is a root, then is a real quadratic factor:
This is always a quadratic with real, positive discriminant-free coefficients.
Finding a cubic with real coefficients given one complex root.
Suppose a cubic polynomial with real coefficients has roots including .
Step 1: By the conjugate root theorem, is also a root.
Step 2: Form the quadratic factor:
Step 3: Since the cubic has degree 3 and we have a quadratic factor, the remaining root must be real. If told the third root is :
Verification: All coefficients are real. ✓
A common mistake is to forget the conjugate root and try to build a cubic with only two roots. Always account for all roots of a degree- polynomial.