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Chapter 8: Problem 37

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ \left(y^{3}-1\right)^{20} $$

Short Answer

Expert verified
The first three terms of the binomial expansion for \( \left(y^{3}-1\right)^{20} \) are \( y^{60}, -20y^{57}, 190y^{54} \).

Step by step solution

01

Understand the binomial theorem

The binomial theorem is \( (a+b)^n = 鈭慱{k=0}^{n} {n \choose k} a^{n-k} b^k \) where \({n \choose k}\) is a binomial coefficient. The binomial coefficient can be calculated using the formula: \({n \choose k} = \frac{n!}{k!(n-k)!}\), where '!' denotes the factorial of a number. Here, a is \(y^3\), b is -1, and n is 20.
02

Find the first term

Substitute k = 0 into the binomial theorem to find the first term. So, the first term is \( {20 \choose 0}(y^{3})^{20-0}(-1)^0 = y^{60} \).
03

Find the second term

Substitute k = 1 into the binomial theorem to find the second term. So, the second term is \( {20 \choose 1}(y^{3})^{20-1}(-1)^1 = -20(y^3)^{19} = -20y^{57} \).
04

Find the third term

Substitute k = 2 into the binomial theorem to find the third term. So, the third term is \( {20 \choose 2}(y^{3})^{20-2}(-1)^2 = {20 \choose 2} y^{54} = 190y^{54} \).
05

Combine the first three terms

Combine the three calculated terms to gain the first three terms of the expansion. Therefore, the first three terms of the binomial expansion are \( y^{60} - 20y^{57} + 190y^{54} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Theorem
The binomial theorem is a fundamental tool in algebra used for expanding expressions that are raised to a power. This theorem provides a way to express these expressions, written as
  • \( (a+b)^n \)
as a sum involving terms of the form
  • \( {n \choose k} a^{n-k} b^k \), where \( k \) denotes a particular term in the expansion
Each term in the expression is affected by both its position \( k \) and the total power \( n \).
In our exercise, for the expression \( (y^3 - 1)^{20} \), the theorem is used to determine the first few terms by setting
  • \( a = y^3 \)
  • \( b = -1 \)
  • \( n = 20 \)
This approach helps in quickly expanding the expression without having to multiply it out fully, making calculations shorter and more manageable. The key is using the theorem efficiently to compute and simplify the terms correctly.
Binomial Coefficient
The binomial coefficient is a central part of the binomial theorem. Represented by \( {n \choose k} \), it tells you how many ways you can choose \( k \) items from \( n \) items, without regard to order. The formula for calculating it is given by:
  • \( {n \choose k} = \frac{n!}{k!(n-k)!} \)
The binomial coefficient plays a critical role in determining the size of each term in the binomial expansion.
In the example \( (y^3 - 1)^{20} \), each coefficient is calculated by plugging \( n = 20 \) into the formula and varying \( k \) to figure out the coefficients for each expanding term.
  • For \( k = 0 \), \( {20 \choose 0} = 1 \)
  • For \( k = 1 \), \( {20 \choose 1} = 20 \)
  • For \( k = 2 \), \( {20 \choose 2} = 190 \)
Understanding how to calculate and apply these coefficients is essential for any binomial expansion.
Factorial
Factorials are a mathematical concept that are crucial for calculating binomial coefficients. Notated as \( n! \), the factorial of a number \( n \) is defined as the product of all positive integers up to \( n \). For example,
  • \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \)
Factorials appear in the binomial coefficient formula \( {n \choose k} = \frac{n!}{k!(n-k)!} \), simplifying the computation of combinations necessary for binomial expansions.
In our example of \( (y^3 - 1)^{20} \), factorials were used to calculate the coefficients:
  • \( {20 \choose 1} = \frac{20!}{1! \times 19!} = 20 \)
  • \( {20 \choose 2} = \frac{20!}{2! \times 18!} = 190 \)
These computations highlight how factorials contribute to solving binomial expansion problems efficiently.

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Most popular questions from this chapter

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