Math Problem Statement

Let f(x) be a polynomial function. If one of the zeros of f(x) is 4i, what is another zero of f(x)?

Solution

In the question provided, you are given that one of the zeros of the polynomial function f(x)f(x) is 4i4i, and you are asked to find another zero.

Key concept:

For polynomial functions with real coefficients, complex roots always occur in conjugate pairs. This means that if 4i4i is a zero, its conjugate, 4i-4i, must also be a zero of f(x)f(x).

Answer:

Another zero of f(x)f(x) is 4i-4i.

Would you like more details on this concept, or do you have any questions? Here are some related questions to explore:

  1. What are complex conjugates and why do they occur in pairs for polynomials with real coefficients?
  2. How would you find a polynomial if the given zeros include complex numbers like 4i4i and 4i-4i?
  3. What is the fundamental theorem of algebra, and how does it relate to the number of zeros a polynomial can have?
  4. How do you express a polynomial with complex roots in standard form?
  5. Can a polynomial have non-real complex roots if all its coefficients are real?

Tip: Always remember that the complex conjugate root theorem applies to polynomials with real coefficients, ensuring symmetry in their complex roots.

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Complex Numbers
Complex Conjugates

Formulas

Complex Conjugate: If a + bi is a root, then a - bi is also a root in polynomials with real coefficients

Theorems

Complex Conjugate Root Theorem
Fundamental Theorem of Algebra

Suitable Grade Level

Grades 10-12