Problem 37 - College Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the binomial theorem

The binomial theorem is \( (a+b)^n = ∑_{k=0}^{n} {n \choose k} a^{n-k} b^k \) 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.

Step 2: Find the first term

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

Step 3: Find the second term

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

Step 4: Find the third term

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

Step 5: 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 \( y^{60} - 20y^{57} + 190y^{54} \).

Key Concepts

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.