Complex Conjugate Theorem

Definition: The Complex Conjugate Theorem states that if a polynomial has real coefficients, then any complex zeros occur in conjugate pairs. This tells us that if a + bi is a zero, then so is a – bi and vice-versa.

Example: If the polynomial x³ – 3x² + 9x + 13 has a zero of 2 – 3i, use the Complex Conjugate Theorem to name another zero.

Solution: 

Step 1: The theorem tells us that as long as the polynomial has real coefficients, then we can simply state the conjugate of our given zero as another zero. The coefficients of this polynomial are real numbers, so we can apply this theorem.

Step 2: The conjugate of 2 – 3i is 2 + 3i, so we know 2 + 3i is also a zero.

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