Question

choose the answer that best completes the sentence.

if you know a root of a function is (-2 + sqrt{5}i), then ______.

• (2 + sqrt{5}i) is a possible root.
• (2 - sqrt{5}i) is a known root.
• (-2 - sqrt{5}i) is a possible root.
• (-2 - sqrt{5}i) is a known root.

Explanation:

🆕 New Concept Discovered: Complex Conjugate Theorem
Irrational and complex roots of polynomials with rational coefficients always come in conjugate pairs.

Step 1: Identify the given root

The problem states that a known root of a polynomial function is:

$$ -2 + \sqrt{3}i $$

This is a complex number in the form \( a + bi \), where \( a = -2 \) and \( b = \sqrt{3} \).

Step 2: Apply the Complex Conjugate Theorem

The Complex Conjugate Theorem states that if a polynomial has real coefficients and a complex number \( a + bi \) is a root, then its complex conjugate \( a - bi \) must also be a root of the function.

To find the conjugate, we change the sign of the imaginary part:

$$ -2 - \sqrt{3}i $$

Since the theorem guarantees this conjugate is a root, it is a known root, not just a possible one.

Answer:

\(-2 - \sqrt{3}i\) is a known root.