Problem 54 - College Algebra

Question

Describe the pattern on the exponents on \(b\) in the expansion of \((a+b)^{n}\).

Step-by-Step Solution

Answer

The exponents on \(b\) in the expansion of \((a+b)^{n}\) start from 0 and increase by 1 in each term until it reaches \(n\).

Step 1: Understand the binomial theorem

The binomial theorem allows the expansion of an expression of the form \((a+b)^n\). The general term of the expansion is given by \(_nC_r a^{n-r} b^r\), where \(_nC_r\) denotes the number of ways to choose \(r\) elements from \(n\), \(a^{n-r}\) denotes \(a\) raised to the power \(n-r\), and \(b^r\) denotes \(b\) raised to the power \(r\). Here, \(r\) is an integer from 0 to \(n\).

Step 2: Identify the exponent on \(b\) in the general term

In the general term \(_nC_r a^{n-r} b^r\) of the binomial expansion, the exponent on \(b\) is \(r\). According to the binomial theorem, \(r\) can have values from 0 to \(n\). Thus, in the expanded form, the exponent on \(b\) can have any value from 0 to \(n\).

Step 3: Describe the pattern

The pattern on the exponents in each term in the expansion of \((a+b)^n\) is that the exponent on \(b\) starts at 0 in the first term, increases by 1 in each subsequent term, and goes up to \(n\) in the last term of the expansion. Each increase in the exponent of \(b\) is typically paired with a corresponding decrease in the exponent of \(a\), such that the sum of the exponents in any given term always equals \(n\).

Key Concepts

Binomial Expansion

When we talk about binomial expansion, we're referring to the process of expanding a binomial expression raised to a power, such as \[(a+b)^n.\] The binomial theorem provides a neat way of finding the expanded form without any hassle. Each term in this expansion follows a specific format, expressed by the general term:

• \(_nC_r\) is the binomial coefficient which tells us how many ways we can choose \(r\) elements from a set of \(n\) elements.
• \(a^{n-r}\) and \(b^r\) indicate the respective powers of \(a\) and \(b\) observed in each term of the expansion.

These general terms combine to form the full expanded expression, which includes terms such as \(a^n\), \(a^{n-1}b\), all the way to \(b^n\). The beauty of the binomial theorem is that it provides a blueprint for any expression of the form \((a+b)^n\). This makes calculations more straightforward, especially for large \(n\).

Exponents

The role of exponents in the binomial expansion is crucial, as they dictate the power to which each component, \(a\) or \(b\), is raised in every term. In the expression of the form \( (a+b)^n \), the calculation of each term involves raising \(a\) and \(b\) to certain powers. Specifically, in the term \(_nC_ra^{n-r}b^r\):

• The exponent on \(a\) is \(n-r\), starting from \(n\) and decreasing with each successive term.
• The exponent on \(b\) begins at 0 and increases up to \(n\), as \(r\) increases from 0 to \(n\).

The summation of the exponents of \(a\) and \(b\) in each term is always equal to \(n\). This uniform change in exponents reflects the underlying pattern of the binomial expansion.

Polynomial Expressions

Polynomial expressions are expressions consisting of terms with varying powers and coefficients. When expanding a binomial like \((a+b)^n\), the result is a polynomial expression. This expansion transforms a two-term expression into multiple terms, where each term is crafted by the general binomial term formula:

• Terms include components like \(a^2b\) or \(ab^2\), each with its corresponding coefficient.
• Term arrangement follows the sequence determined by the exponents \(n-r\) and \(r\).

Within the polynomial, the arrangement showcases a systematic pattern of increasing powers of \(b\) and decreasing powers of \(a\). This kind of pattern makes polynomial expressions derived from binomial expansions predictable and manageable.

Combinatorics

Combinatorics plays a fundamental role in understanding binomial expansions, particularly through its influence on binomial coefficients. These coefficients resemble the counting principles used to select combinations of items from a larger pool. In the term \(_nC_ra^{n-r}b^r\):

• \(_nC_r\) is calculated using the formula \(\frac{n!}{r!(n-r)!}\), where \(!\) denotes factorial, representing the number of ways of choosing \(r\) elements from \(n\).
• This coefficient reflects the frequency with which each term appears in the expansion.

Understanding combinatorics helps in appreciating the symmetrical properties and numerical relationships within the terms of a binomial expansion. This insight connects mathematical theory with practical counting and arrangement scenarios of everyday contexts.