Math Problem Statement

Information is given about a polynomial f(x) whose coefficients are real numbers. Find the remaining zeros of f. Degree 3; zeros: -1, 8 - i

Solution

To solve this problem, we need to determine the remaining zeros of a polynomial f(x)f(x) of degree 3 with real coefficients, given that two of its zeros are 1-1 and 8i8 - i.

Key Points:

  1. Complex Conjugate Root Theorem: If a polynomial with real coefficients has a complex root, then the conjugate of that root must also be a root.

  2. Applying the Conjugate:

    • Since 8i8 - i is a zero, its conjugate 8+i8 + i must also be a zero to satisfy the requirement of real coefficients.
  3. Degree Check:

    • The polynomial is of degree 3, so it must have exactly three roots. We already have:
      • 1-1
      • 8i8 - i
      • 8+i8 + i

Thus, the remaining zero of f(x)f(x) is 8+i8 + i.

Answer:

The remaining zero of f(x)f(x) is: 8+i8 + i

Would you like more detailed explanations on complex roots or the Complex Conjugate Root Theorem?


Related Questions:

  1. How do we determine if a polynomial with real coefficients will have complex roots?
  2. What is the Complex Conjugate Root Theorem, and why is it important?
  3. Can a polynomial of degree 4 have exactly one real root?
  4. How do we construct a polynomial when given its roots?
  5. What are some real-world applications of polynomials with complex roots?

Tip: For any polynomial with real coefficients, remember that complex roots always appear in conjugate pairs.

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomials
Complex Numbers

Formulas

-

Theorems

Complex Conjugate Root Theorem

Suitable Grade Level

Grades 10-12