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Chapter 14: Problem 38

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

Short Answer

Expert verified
The first three terms in the expansion are: \(y^{63}\), \(-21y^{60}\), and \(-105y^{57}\).

Step by step solution

01

Identify the terms

In this exercise the binomial is \((y^{3}-1)\). So \(a = y^{3}\) and \(b = -1\). These are the two terms we will use in the binomial expansion.
02

First Term Calculation

To find the first term in the expansion use the binomial theorem, where it is \(a^n\). So the first term is \(y^{3 \times 21} = y^{63}\).
03

Second term calculation

For the second term, the binomial theorem formula gives it as \(na^{n-1}b\), substituting the terms, \(21 \times (y^{3 \times 20}) \times -1 = -21y^{60}\).
04

Third term calculation

For the third term, the formula gives \(n(n-1)a^{n-2}b^2/2!\). Substituting in, results in \(-210y^{57}\). Dividing by 2! which is 2, we get \(-105y^{57}\).

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