Complex conjugate roots
Objective. Use conjugate pairing for real-coefficient polynomials.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
For a polynomial with real coefficients, any non-real root must occur together with its complex conjugate, since the coefficients are unaffected by taking conjugates throughout the equation. This means non-real roots always come in conjugate pairs, so a cubic with real coefficients has either three real roots or one real root and a conjugate pair. For example, if 2 + 3i is a root of a real cubic, then 2 − 3i must also be a root, and their product (2+3i)(2−3i)=13 gives a real quadratic factor x²−4x+13.
Worked example
A cubic with real coefficients has roots 3 and 1 + 2i. Find the third root and the cubic equation in the form x³+bx²+cx+d=0.
- 1Since coefficients are real, 1 − 2i must also be a root
- 2The quadratic factor from 1+2i and 1−2i is (x−1)²+4 = x² − 2x + 5
- 3Multiply by the factor (x−3): (x−3)(x²−2x+5)
- 4Expand: x³ − 2x² + 5x − 3x² + 6x − 15 = x³ − 5x² + 11x − 15
Answer: Third root 1 − 2i; cubic x³ − 5x² + 11x − 15 = 0
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Practice
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Start practisingCommon mistake. Students assume complex roots pair up even when the polynomial has non-real coefficients. Conjugate pairing only applies when all coefficients of the polynomial are real; always check this condition first.
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