Math Problem Statement

A polynomial equation with rational coefficients has the roots 3 + √4, 3 - √6. Find two additional roots.

Solution

To determine the additional roots of the polynomial with rational coefficients:

Key Concept:

If a polynomial has rational coefficients and contains irrational roots like 3+43 + \sqrt{4} or 363 - \sqrt{6}, their conjugates 343 - \sqrt{4} and 3+63 + \sqrt{6} must also be roots of the polynomial.

Given Roots:

  • 3+43 + \sqrt{4}
  • 363 - \sqrt{6}

Additional Roots:

Their conjugates are:

  • 343 - \sqrt{4}
  • 3+63 + \sqrt{6}

Correct Answer:

The correct option is:

  • 34,3+63 - \sqrt{4}, 3 + \sqrt{6}

Let me know if you'd like a detailed explanation or further assistance.


5 Related Questions:

  1. How are conjugate roots determined for polynomials with rational coefficients?
  2. Why must irrational roots occur in conjugate pairs in these cases?
  3. Can a polynomial have one irrational root without its conjugate?
  4. What are examples of polynomials where roots include complex numbers?
  5. How does the degree of a polynomial relate to its number of roots?

Tip: Always ensure the polynomial's coefficients are rational before assuming conjugate roots must exist.

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Math Problem Analysis

Mathematical Concepts

Polynomial equations
Conjugate roots
Rational coefficients

Formulas

-

Theorems

Conjugate root theorem

Suitable Grade Level

Grades 10-12