Problem 70 - College Algebra

Question

Which one of the following is true? a. The binomial expansion for \((a+b)^{n}\) contains \(n\) terms. b. The Binomial Theorem can be written in condensed form as \((a+b)^{n}=\sum_{r=0}^{n}\left(\begin{array}{l}n \\ r\end{array}\right) a^{n-r} b^{r}\). c. The sum of the binomial coefficients in \((a+b)^{n}\) cannot be \(2^{n}\). d. There are no values of \(a\) and \(b\) such that \((a+b)^{4}=a^{4}+b^{4}\)

Step-by-Step Solution

Answer

Out of the four statements, both statement b and statement d are true.

Step 1: Analyze each statement individually

Consider each of the given statements and compare it to what is known about the Binomial Theorem and binomial expansion.

Step 2: Evaluate statement a

The first statement says that the binomial expansion of \((a+b)^{n}\) contains \(n\) terms. However, this is incorrect. The binomial expansion for \((a+b)^{n}\) actually contains \(n+1\) terms.

Step 3: Evaluate statement b

The statement describes the condensed form of the Binomial Theorem as \((a+b)^{n}=\sum_{r=0}^{n}{n \choose r} a^{n-r}b^{r}\). This is correct. It accurately presents the Binomial Theorem.

Step 4: Evaluate statement c

The third statement claims that the sum of the binomial coefficients in \((a+b)^{n}\) cannot be \(2^{n}\). Actually, the sum of the binomial coefficients in \((a+b)^{n}\) is indeed \(2^{n}\). Therefore, this statement is false.

Step 5: Evaluate statement d

The fourth statement states that there are no values of \(a\) and \(b\) such that \((a+b)^{4}=a^{4}+b^{4}\). This statement is true. By the Binomial Theorem, \((a+b)^4 = a^4+4a^3b+6a^2b^2+4ab^3+b^4\). Even if \(a\) or \(b\) are zero, the expanded form is not equal to \(a^{4}+b^{4}\).

Key Concepts

Binomial Expansion

The binomial expansion is a method to expand expressions that are raised to a power. It specifically applies when the expression is in the form of \((a+b)^n\). This expansion results in a series of terms where each term represents a combination of the variables \(a\) and \(b\) raised to various powers. The number of terms in this expansion for \((a+b)^n\) is actually \(n+1\), not \(n\), as it might seem intuitively at first.

• For example, the expansion of \((a+b)^2\) results in \(a^2 + 2ab + b^2\), which contains 3 terms.
• Every term in this expansion is associated with a binomial coefficient, which will be explored further in the next section.

To summarize, binomial expansion uses both powers of \(a\) and \(b\) and coefficients to form a polynomial expression. Ensuring an understanding of this process is essential for solving related algebraic problems.

Binomial Coefficients

Binomial coefficients are an integral part of the binomial expansion. They represent the specific weights or scaling factors for each term of the expansion. These coefficients are represented by the notation \(\binom{n}{r}\), which is called a "combination" and reads as "n choose r".

• The value \(\binom{n}{r}\) indicates the number of ways to choose \(r\) items from a set of \(n\) items without regard to order.
• Mathematically, it is defined as \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(!\) denotes factorial, the product of all positive integers up to that number.
• Each term in the binomial expansion has a binomial coefficient; for instance, in \((a+b)^3\), the expansion \(a^3 + 3a^2b + 3ab^2 + b^3\) has coefficients 1, 3, 3, and 1 respectively.

Understanding these coefficients is crucial as they tell us how many times each power combination of \(a\) and \(b\) appears in the expanded form.

Pascal's Triangle

Pascal's Triangle is a geometric representation that provides a quick and easy way to access binomial coefficients. Each row in Pascal's Triangle corresponds to the coefficients applicable for the expansion of \((a+b)^n\).

• To create Pascal's Triangle, start with a single number 1 at the top (the zeroth row), forming a triangle.
• Each subsequent row starts and ends with 1. Every interior number of a row is found by adding the two numbers directly above it.
• For example, the third row starting with 1 goes \(1, 3, 3, 1\), which suits the coefficients of \((a+b)^3\) expansion.

Pascal's Triangle is a helpful tool for quickly finding binomial coefficients without having to calculate each combination manually. As a result, it simplifies the process of expanding binomials and provides a visual aid to support understanding.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating these symbols. In the context of the binomial theorem and expansion, algebra plays a pivotal role in managing expressions and solving equations.

• It uses letters like \(a\) and \(b\) to represent numbers, which can be manipulated according to set mathematical rules.
• Algebraic methods are essential for deriving conclusions about unknown quantities, and the binomial theorem is one application of these techniques.
• Working through binomial expansion problems often requires skills in factoring, simplifying expressions, and understanding polynomial structures.

Overall, a strong grasp of algebraic principles is necessary to effectively work through problems involving binomial expansions and related computations. This understanding supports critical thinking and problem-solving skills essential for mathematics and its applications.