SUMMARY
The polynomial equation $$x^3-9x+9=0$$ has roots $p$, $q$, and $r$. It is demonstrated that $$p^2+p-6$$ is a root of this cubic equation, distinct from $p$. By substituting $$p^2+p-6$$ into the polynomial, it is confirmed that this expression equals either $q$ or $r$, as neither $\pm \sqrt{6}$ are roots of the cubic. This establishes a definitive relationship between the roots of the polynomial and the expression derived from $p$.
PREREQUISITES
- Understanding of polynomial equations and their roots
- Familiarity with cubic functions and their properties
- Knowledge of substitution methods in algebra
- Basic grasp of real number properties and root distinctions
NEXT STEPS
- Study the properties of cubic polynomials, specifically the Rational Root Theorem
- Explore polynomial root-finding techniques, including synthetic division
- Learn about Vieta's formulas and their applications in polynomial equations
- Investigate the implications of distinct real roots in polynomial functions
USEFUL FOR
Mathematicians, algebra students, and anyone interested in polynomial equations and their roots will benefit from this discussion.