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Magazine Chapter 14: Problem 33
Find the indicated term. The eighth term of the expansion of \((2 c+d)^{7}\)
Short Answer
Expert verified
The eighth term is \(d^7\).
Step by step solution
01
Understand the Binomial Theorem
The Binomial Theorem can be used to expand expressions of the form \((a+b)^n\). It states: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In this case, \((2c + d)^7\), we have \(a = 2c\), \(b = d\), and \(n = 7\).
02
Identify the Term You Need
The general term of a binomial expansion is given by \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\). For the eighth term, we have \(k = 7\) because we start counting from \(k = 0\). Thus we are looking for \(T_{8}\).
03
Substitute Values into the General Term Formula
For \(T_8\), we substitute \(n = 7\), \(k = 7\), \(a = 2c\), and \(b = d\) into the formula \(T_{k+1} = \binom{n}{k} a^{n-k} b^k\). Therefore, we have:\[T_8 = \binom{7}{7} (2c)^{7-7} d^7\].
04
Calculate the Binomial Coefficient
Calculate \(\binom{7}{7}\), which is 1 because it represents the number of ways to choose 7 items out of 7. Thus, \(\binom{7}{7} = 1\).
05
Evaluate the Powers
Now, evaluate the terms: \((2c)^{0} = 1\) because any number to the power of 0 is 1, and \(d^7 = d^7\).
06
Calculate the Eighth Term
Now combine all elements from the previous steps: \[T_8 = 1 \times 1 \times d^7 = d^7\]. Therefore, the eighth term in the expansion is \(d^7\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Coefficient
The binomial coefficient is a crucial part of the Binomial Theorem. It is represented as \( \binom{n}{k} \), which is pronounced as "n choose k". This coefficient tells us the number of ways to select a subset of \(k\) elements from a larger set of \(n\) elements, ignoring the order of selection. In simpler terms:
- If you have 5 shirts and want to choose 2 distinct shirts to wear for the week, the possible combinations are calculated using the binomial coefficient.
- The formula for the binomial coefficient is given by: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
- In factorial terms, \( n! \) means "n factorial", which is the product of all positive integers up to \(n\).
Polynomial Expansion
Polynomial expansion involves expanding an expression elevated to a power into a sum of simpler terms. The Binomial Theorem is a powerful tool for this, as it lets us expand expressions like \((a+b)^n\) efficiently:
- The theorem provides a formula for expansion, which is summarized as: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).
- Each term in the expansion is a combination of powers of \(a\) and \(b\), multiplied by a specific binomial coefficient.
Algebraic Expression
An algebraic expression is a combination of numbers, variables, and operation symbols (like \( +, -, * \)). They form the basis of algebra and are used to model a wide range of problems:
- Expressions consist of terms, which are separated by addition or subtraction operators.
- Variables like \(c\) and \(d\) represent unknowns or quantities that can change value.
- Numerical coefficients (like 2 in \(2c\)) are the numbers multiplying the variables in a term.
Exponentiation
Exponentiation refers to the operation of raising a number, known as the base, to a power, indicating how many times the base is multiplied by itself:
- Mathematically, this is shown as \(a^n\), where \(a\) is the base and \(n\) is the exponent.
- In expressions like \((2c + d)^7\), exponentiation lets us raise the entire expression to the 7th power.
- The power tells us that the base (in this case, the full expression \((2c + d)\)) is multiplied by itself 7 times.
