Problem 38 - Algebra and Trigonometry

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(y^{3}-1\right)^{21} $$

Step-by-Step Solution

Answer

The first three terms in the binomial expansion of \(\left(y^{3}-1\right)^{21}\) are \(y^{63}-21y^{60}+210y^{57}\)

Step 1: Identify the components of the binomial

The binomial is \(y^{3}-1\) and it is raised to power 21. So, A = \(y^3\), B = -1 and n = 21.

Step 2: Calculate the first term of the expansion

Using the formula for the rth term of a binomial expansion, we get T_1 = ^21C_0 * \((y^{3})^{21}\) * (-1)^0= 1 * \(y^{63}\) * 1 = \(y^{63}\).

Step 3: Calculate the second term of the expansion

For T_2, r = 1. Substituting these values into the formula, T_2= ^21C_1 * \((y^{3})^{20}\) * (-1)^1= -21*(\(y^{60}\)) = -21\(y^{60}\).

Step 4: Calculate the third term of the expansion

For T_3, r = 2. Substituting these values into the formula, T_3= ^21C_2 * \((y^{3})^{19}\) * (-1)^2= 210 * \(y^{57}\) = 210\(y^{57}\).