SUMMARY
The integers p and q for which \( x = \sqrt{29} + \sqrt{89} \) is a root of the equation \( x^4 + px^2 + q = 0 \) are determined through substitution and simplification. By calculating \( x^2 \), we find \( x^2 = 29 + 89 + 2\sqrt{29 \cdot 89} = 118 + 2\sqrt{2581} \). Subsequently, substituting \( x^2 \) into the polynomial reveals that \( p = -118 \) and \( q = -2581 \). Thus, the solution is \( p = -118 \) and \( q = -2581 \).
PREREQUISITES
- Understanding of polynomial equations and roots
- Knowledge of square roots and their properties
- Familiarity with algebraic manipulation and simplification
- Basic concepts of integer solutions in equations
NEXT STEPS
- Study polynomial root-finding techniques
- Explore the properties of square roots in algebra
- Learn about the Rational Root Theorem
- Investigate integer solutions in higher-degree polynomials
USEFUL FOR
Mathematicians, algebra students, and educators looking to deepen their understanding of polynomial equations and integer solutions.