LEARN · EXPLAIN · REVISE

Explain why the conjugate of a complex root is also a root of a polynomial with real coefficients.

Read the idea, work independently, then explain what changed.

高一必修第二册(A版).pdf · 7.2 · PDF 82 / printed page 75

TOPIC 01

Arithmetic of complex numbers

Build understanding of arithmetic of complex numbers through definitions, contrasting cases and justified applications.

Use one hint at a time. A correction explains what changed, not just the final answer.

01 / Standard#Your turn

p(z)=0,\quad p\in\mathbb R[z]

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Conjugation preserves sums and products.

Hint 2

Real coefficients are unchanged by conjugation.

Worked solution

Conjugate every term: p(conjugate z)=conjugate p(z)=0.

Checks and common pitfalls: The real-coefficient assumption is essential.

Reasoning checklist · self / teacher assessment

Think first. Reveal a hint when the class is ready.

Teacher preparation and assessment

Connect geometric and algebraic forms of arithmetic of complex numbers.
Multiplication: Expand algebraically and replace i² with −1.
Division: A conjugate turns a nonzero complex denominator into its positive squared modulus.
Close with: conditions → representation → reasoning → check.

Expected reasoning: Expand algebraically and replace i² with −1.
Expected reasoning: A conjugate turns a nonzero complex denominator into its positive squared modulus.
Expected correction: The sign from i² must not be lost.