Complex roots come in pairs
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Flip to reveal answersIf a real-coefficient polynomial has root a + bi, what else is a root?
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All 8 Flashcards — Complex roots come in pairs
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Question
If a real-coefficient polynomial has root a + bi, what else is a root?
Answer
Its conjugate a − bi — complex roots come in conjugate pairs.
Question
What real quadratic has roots a ± bi?
Answer
z² − 2az + (a² + b²) (middle term −sum, constant = product).
Question
How do you finish a polynomial given one complex root?
Answer
Write the conjugate, form their real quadratic, divide it out, then solve what's left.
Question
Why does the conjugate-pair rule need real coefficients?
Answer
The proof relies on conjugating the whole equation; with real coefficients the equation is unchanged, forcing the conjugate to be a root.
Question
Another root if 2 + i is a root (real coefficients)?
Answer
2 − i.
Question
Real quadratic with roots 1 ± 2i?
Answer
z² − 2z + 5 (sum 2, product 5).
Question
Can a real cubic have exactly two real roots and one complex root?
Answer
No — complex roots come in pairs, so a cubic has either 3 real roots or 1 real + a conjugate pair.
Question
Roots of z³ − 3z² + 7z − 5 given 1 + 2i is one?
Answer
1 + 2i, 1 − 2i, 1.
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Full study notes for Complex roots come in pairs
Topic 1.14 hub
De Moivre & roots (HL only)
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