Problem 21
Question
What are the first and last terms in the expansion of \((2 x+3 y)^{4} ?\)
Step-by-Step Solution
Verified Answer
The first term is \(16x^4\) and the last term is \(81y^4\).
1Step 1: Understand the Binomial Theorem
The binomial theorem states that: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\), where \(\binom{n}{k}\) is a binomial coefficient. It helps in expanding an expression of the form \((a + b)^n\).
2Step 2: Identify the Given Terms
In the expression \((2x+3y)^4\), we have \(a = 2x\), \(b = 3y\), and \(n = 4\).
3Step 3: Find the First Term
The first term in the expansion corresponds to \(k = 0\) in the binomial expansions. It is given by \(\binom{4}{0} (2x)^{4} (3y)^{0}\). Evaluate: \(\binom{4}{0} = 1\), \((2x)^4 = 16x^4\), \((3y)^0 = 1\). Multiplying these gives the first term: \(16x^4\).
4Step 4: Find the Last term
The last term in the expansion corresponds to \(k = 4\) in the binomial expansion. It is given by \(\binom{4}{4} (2x)^{0} (3y)^{4}\). Evaluate: \(\binom{4}{4} = 1\), \((2x)^0 = 1\), and \((3y)^4 = 81y^4\). Multiplying these gives the last term: \(81y^4\).
Key Concepts
Binomial ExpansionBinomial CoefficientAlgebraic Expressions
Binomial Expansion
The binomial expansion is a method used to expand expressions of the form \((a + b)^n\) into a sum of terms involving powers of both \(a\) and \(b\). This powerful algebraic technique is derived from the Binomial Theorem, introducing a systematic way of finding the expanded form for binomials raised to any positive integer \(n\).
The full expansion results in terms that include the binomial coefficients, which represent the number of ways of selecting elements from a set. Each term in the expansion also involves powers of the variables.
For instance, considering the expression \((2x+3y)^4\), the expansion will produce five terms. However, students often focus on deriving specific terms like the first and last, which are easier to calculate directly using the properties of the binomial expansion.
Moreover, understanding binomial expansion is crucial in simplifying many algebraic expressions, as it provides a structured approach to expanding even higher-order powers of binomials.
The full expansion results in terms that include the binomial coefficients, which represent the number of ways of selecting elements from a set. Each term in the expansion also involves powers of the variables.
For instance, considering the expression \((2x+3y)^4\), the expansion will produce five terms. However, students often focus on deriving specific terms like the first and last, which are easier to calculate directly using the properties of the binomial expansion.
Moreover, understanding binomial expansion is crucial in simplifying many algebraic expressions, as it provides a structured approach to expanding even higher-order powers of binomials.
Binomial Coefficient
The binomial coefficient, commonly represented as \(\binom{n}{k}\), plays a pivotal role in binomial expansion. It is calculated using the formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]where \(n!\) denotes the factorial of \(n\), which is the product of all positive integers from 1 to \(n\). The binomial coefficient calculates the number of ways to choose \(k\) elements from a set of \(n\) elements without regard to order.
In the context of the expression \((2x+3y)^4\), the coefficients help determine the multiplier for each term in the expansion. For example:- The first term uses \(\binom{4}{0}\), resulting in the coefficient 1.- The last term uses \(\binom{4}{4}\), also resulting in the coefficient 1. Each term in the expansion requires calculating a different binomial coefficient, dictating how the power of each variable affects the expansion. Calculating these coefficients correctly is essential in solving binomial expansion problems effectively and accurately.
In the context of the expression \((2x+3y)^4\), the coefficients help determine the multiplier for each term in the expansion. For example:- The first term uses \(\binom{4}{0}\), resulting in the coefficient 1.- The last term uses \(\binom{4}{4}\), also resulting in the coefficient 1. Each term in the expansion requires calculating a different binomial coefficient, dictating how the power of each variable affects the expansion. Calculating these coefficients correctly is essential in solving binomial expansion problems effectively and accurately.
Algebraic Expressions
Algebraic expressions are fundamental components in mathematics, composed of variables, constants, and arithmetic operations. They represent quantities and relationships between quantities, making them essential in problem-solving and logical reasoning.
In problems like \((2x+3y)^4\), algebraic expressions are expanded to form new expressions that make calculations easier or provide insight into various properties of the expression.
During the expansion, elements like \(2x\) and \(3y\) are treated as single units (or "terms") that undergo operations specified by the binomial theorem. Thus, understanding how to manipulate algebraic expressions is key to mastering expansion techniques and simplifying complex problems.
By working through expansions like this, students gain a deeper understanding of not only algebraic manipulation but also the dynamics of variable interactions within an expression. This skill is crucial for advanced mathematical topics, including polynomial identities and calculus.
In problems like \((2x+3y)^4\), algebraic expressions are expanded to form new expressions that make calculations easier or provide insight into various properties of the expression.
During the expansion, elements like \(2x\) and \(3y\) are treated as single units (or "terms") that undergo operations specified by the binomial theorem. Thus, understanding how to manipulate algebraic expressions is key to mastering expansion techniques and simplifying complex problems.
By working through expansions like this, students gain a deeper understanding of not only algebraic manipulation but also the dynamics of variable interactions within an expression. This skill is crucial for advanced mathematical topics, including polynomial identities and calculus.
Other exercises in this chapter
Problem 21
Find \(a_{8}\) and \(a_{n}\) for each arithmetic sequence. $$a_{1}=x, a_{2}=x+3$$
View solution Problem 21
Evaluate each expression. Do not use a calculator. $$C(6,0)$$
View solution Problem 21
Find \(a_{1}\) and \(r\) for each geometric sequence. $$a_{3}=5, a_{8}=\frac{1}{625}$$
View solution Problem 21
Decide whether each sequence is finite or infinite. $$a_{1}=3 ; \text { for } 2 \leq n \leq 10, a_{n}=3 \cdot a_{n-1}$$
View solution