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Magazine Chapter 13: Problem 48
Find the indicated term of each binomial expansion. $(r-4)^{11} ; fourth term
Short Answer
Expert verified
The fourth term is \ -10560 r^8 \.
Step by step solution
01
Identify the binomial and exponent
The binomial is \(r - 4\) and the exponent is 11. We are asked to find the fourth term in the expansion of \((r-4)^{11}\).
02
Use the binomial expansion formula
The general term in the expansion of \( (a + b)^n \) is given by \[ T_{k+1} = \binom{n}{k} a^{n-k} b^k \] For the given problem, \(a = r \), \( b = -4 \) and \( n = 11 \).
03
Set up the fourth term
To find the fourth term, we use \( k = 3 \) because the fourth term corresponds to \( k+1 = 4 \). So we will find \( T_4 = \binom{11}{3} r^{11-3} (-4)^3 \).
04
Calculate the binomial coefficient
The binomial coefficient is \( \binom{11}{3} = \frac{11!}{3!(11-3)!} = \frac{11!}{3! \cdot 8!} = 165 \).
05
Compute the powers
Compute \(r^{11-3} = r^8 \) and \( (-4)^3 = -64 \).
06
Combine all parts
Multiply the binomial coefficient, \(r^8\), and \(-64\) to get the fourth term: \[ T_4 = 165 \cdot r^8 \cdot (-64) = -10560 r^8 \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Theorem
The binomial theorem is a fundamental principle in algebra that provides a way to expand expressions of the form \( (a + b)^n \). The theorem states that the expansion can be written as:
\[ (a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \text{...} + \binom{n}{n} a^0 b^n \]
Each term in the expansion is determined by the binomial coefficient \( \binom{n}{k} \), the power of the first term (\(a\)), and the power of the second term (\(b\)).
\[ (a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \text{...} + \binom{n}{n} a^0 b^n \]
Each term in the expansion is determined by the binomial coefficient \( \binom{n}{k} \), the power of the first term (\(a\)), and the power of the second term (\(b\)).
- \( a \) and \( b \) are constants or variables in the expression.
- \( n \) is a non-negative integer representing the exponent.
- \( k \) is the term number minus one (e.g., for the fourth term, \( k = 3 \)).
Binomial Coefficient
A binomial coefficient is a number that appears in the expansion of a binomial expression \( (a + b)^n \). It is represented as \( \binom{n}{k} \) and can be calculated using factorials:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, \( n! \) (n factorial) is the product of all positive integers up to \( n \) (i.e., \( n! = n \times (n-1) \times \text{...} \times 1 \)). For example, to find \( \binom{11}{3} \):
Then, \[ \binom{11}{3} = \frac{11!}{3! \times 8!} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165 \]
This coefficient helps in determining the magnitude of each term in the binomial expansion.
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, \( n! \) (n factorial) is the product of all positive integers up to \( n \) (i.e., \( n! = n \times (n-1) \times \text{...} \times 1 \)). For example, to find \( \binom{11}{3} \):
- Compute \( 11! = 11 \times 10 \times 9 \times 8! \).
- Compute \( 3! = 3 \times 2 \times 1 = 6 \).
- Compute \( 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).
Then, \[ \binom{11}{3} = \frac{11!}{3! \times 8!} = \frac{11 \times 10 \times 9}{3 \times 2 \times 1} = 165 \]
This coefficient helps in determining the magnitude of each term in the binomial expansion.
Exponents
Exponents are used to express repeated multiplication of a number by itself. In the context of binomial expansion, they are the powers to which the variables (terms) are raised. For example, in \( (r-4)^{11} \), the exponent 11 indicates that the binomial \( (r-4) \) is multiplied by itself 11 times.
For example, to find the fourth term in the expansion of \( (r-4)^{11} \), we have:
Therefore, the fourth term is calculated as:
\[ T_4 = \binom{11}{3} r^8 (-4)^3 = 165 \times r^8 \times (-64) = -10560 r^8 \]
Exponents are a crucial part of understanding and computing terms in a binomial expansion.
- In each term of the expansion, \( r \) and \( -4 \) will be raised to complementary exponents.
- The powers of \( r \) start at 11 and decrease, while the powers of \( -4 \) start at 0 and increase.
For example, to find the fourth term in the expansion of \( (r-4)^{11} \), we have:
- The exponent of \( r \) will be 11-3 = 8.
- The exponent of \( -4 \) will be 3.
Therefore, the fourth term is calculated as:
\[ T_4 = \binom{11}{3} r^8 (-4)^3 = 165 \times r^8 \times (-64) = -10560 r^8 \]
Exponents are a crucial part of understanding and computing terms in a binomial expansion.
