Complex conjugate root theorem

For a polynomial with real coefficients, every nonreal complex root occurs with its complex conjugate and the same multiplicity.

Complex conjugation: The operation that maps a complex number a + bi to a − bi. Conjugating a polynomial value preserves its coefficients when those coefficients are real.
Complex number: A number expressible as a + bi, where a and b are real and i² = −1. Nonreal roots live in the complex numbers, where conjugation creates the paired value.
Real polynomial factorization: The expression of a real-coefficient polynomial as a product of factors with real coefficients. Each nonreal conjugate pair combines into a real quadratic factor.
x² + 1: The polynomial x² + 1, whose roots are i and −i. Its simplest nonreal roots demonstrate the theorem directly.
Polynomial evaluation: The process of substituting a value into a polynomial to compute its output. Applying conjugation to a polynomial's value shows why a root forces its conjugate to be a root.
Imaginary unit: The complex number i whose square is −1. Its sign changes under conjugation, turning a + bi into a − bi.
Quadratic formula: A formula that gives the roots of a quadratic equation from its coefficients. For real coefficients, its nonreal solutions appear as conjugates.
x² − 2x + 5: The quadratic polynomial with roots 1 + 2i and 1 − 2i. Its roots form a nonreal conjugate pair and its coefficients are real.
Real number: A number on the continuous number line, including rational and irrational numbers. Real coefficients remain unchanged under complex conjugation, which makes the theorem hold.
Root of a polynomial: A value that makes a polynomial evaluate to zero. The theorem describes which other values must also be roots.

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Complex conjugateBroader topic: It is a consequence of how conjugation interacts with polynomial evaluation.