Problem 21 - Algebra and Trigonometry

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ \left(2 x^{3}-1\right)^{4} $$

Step-by-Step Solution

Answer

The expanded form of \((2x^3 - 1)^4\) is \(16x^{12} - 96x^9 + 144x^6 - 96x^3 + 1\).

Step 1: Understanding Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\), where \({n \choose k}\) is the binomial coefficient.

Step 2: Selecting the coefficients

The coefficients for the fourth power are 1, 4, 6, 4, 1, so these will be the coefficients for each term in the expanded binomial.

Step 3: Applying Binomial Theorem

The binomial \((2x^3 - 1)^4\) can be expanded as \(1*(2x^3)^4 + 4*(2x^3)^3*(-1) + 6*(2x^3)^2*(-1)^2 + 4*(2x^3)*(-1)^3 + 1*(-1)^4\).

Step 4: Simplifying Each Term

The previous expression simplifies to \(16x^{12} - 96x^9 + 144x^6 - 96x^3 + 1\).