Problem 38 - College Algebra

Question

In Exercises 31-38, 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 are \(y^{63}, -21y^{60}, 210y^{57}\).

Step 1: Understand Binomial Theorem

The binomial theorem tells us that \( (a+b)^n = \sum_{k=0}^n {n\choose k} a^{n-k} b^k \), where \( {n\choose k} \) denotes binomial coefficients.

Step 2: Identifying the variables in our binomial

In our binomial \( \left(y^{3}-1\right)^{21} \), here \( a = y^{3} \), \( b = -1 \) and \( n = 21 \).

Step 3: Apply the Binomial Theorem

The first three terms of this binomial expansion can be obtained by substituting n, a and b into the formula for the first three terms which are when \( k = 0, 1, 2 \) respectively.

Step 4: Compute first term

For \( k = 0 \), the first term is \( {21\choose 0} * (y^{3})^{21-0} * (-1)^{0} = y^{63} \).

Step 5: Compute second term

For \( k = 1 \), the second term is \( {21\choose 1} * (y^{3})^{21-1} * (-1)^{1} = - 21*y^{60} \).

Step 6: Compute third term

For \( k = 2 \), the third term is \( {21\choose 2} * (y^{3})^{21-2} * (-1)^{2} = 210*y^{57} \).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful and essential mathematical tool used to expand expressions of the form \((a+b)^n\). This theorem allows us to express the powers of a binomial, which is a simple sum of two terms, into a polynomial of terms. The expansion is done as per the formula:\[(a+b)^n = \sum_{k=0}^n {n\choose k} a^{n-k} b^k\]Here \(a\) and \(b\) are any real numbers, and \(n\) is a non-negative integer. The symbol \(\sum\) represents the summation of all the terms from \(k=0\) to \(k=n\). Each term of the expansion involves powers of \(a\) and \(b\), multiplied together and then scaled by a binomial coefficient. Using this theorem, a seemingly complicated expression like \((y^3 - 1)^{21}\) becomes manageable. The theorem helps you calculate the powers systematically without multiplying everything out by hand.

Binomial Coefficient

The binomial coefficient, which appears in the binomial theorem, plays a crucial role in simplifying polynomial expansions. It is denoted by \({n\choose k}\), read as "n choose k," and can be calculated using the formula:\[{n\choose k} = \frac{n!}{k!(n-k)!}\]This formula represents the number of ways to choose \(k\) elements from a set of \(n\) elements without considering the order of selection. In the context of the binomial expansion, the binomial coefficients give the weights by multiplying with each respective term in the expansion, ensuring that all possibilities and permutations are accounted.For example, in our exercise \({21\choose 0}\), \({21\choose 1}\), and \({21\choose 2}\) were calculated as 1, 21, and 210 respectively. The coefficient tells us how many times the basic product of powers appears in the expanded form.

Polynomial Expansion

In mathematics, the polynomial expansion allows us to rewrite expressions involving powers of binomials or any polynomials in a simpler and expanded format. Here, it refers to transforming \( (y^3 - 1)^{21} \) into a sequence of terms involving \( y \) raised to various powers with specific coefficients derived from the binomial theorem.The aim of a polynomial expansion is to break down a complex polynomial into its simpler constituents. Each term in the result consists of a product of powers of singular terms combined with their respective binomial coefficients, making it more manageable and easier to work with.For the polynomial expansion of this example, the first few terms are derived directly from applying the binomial theorem using the identified components: \( a = y^3, b = -1, \) and \( n = 21. \) When expanded, the powers and coefficients allow for a better understanding of how such large expressions "expand out" and behave under various mathematical operations.