SUMMARY
The discussion focuses on solving a problem related to the binomial theorem, specifically the expansion of the expression $$x^2\left(3x^2+\frac{k}{x}\right)^8$$ to find the constant term, which is given as 16128. The relevant term in the expansion is identified as $$3^{8-i}{8 \choose i}k^ix^{18-3i}$$. To determine the value of \( k \), the exponent of \( x \) must equal zero, leading to the equation \( 18 - 3i = 0 \), which simplifies to \( i = 6 \).
PREREQUISITES
- Understanding of the binomial theorem and its application in polynomial expansions.
- Familiarity with combinatorial notation, specifically binomial coefficients.
- Basic algebraic manipulation skills to solve for variables in equations.
- Knowledge of constant terms in polynomial expressions.
NEXT STEPS
- Study the derivation and application of the binomial theorem in polynomial expansions.
- Learn how to calculate binomial coefficients using the formula $$n \choose k$$.
- Explore methods for identifying constant terms in polynomial expressions.
- Practice solving similar problems involving variable coefficients in polynomial expansions.
USEFUL FOR
Students studying algebra, mathematics enthusiasts, and educators looking for examples of the binomial theorem in action.