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Magazine Chapter 9: Problem 22
In Exercises \(18-22,\) use the Binomial Theorem to find the indicated term. The constant term in the expansion \(\left(x+x^{-1}\right)^{8}\)
Short Answer
Expert verified
The constant term is 70.
Step by step solution
01
Identify the General Term
The general term of a binomial expansion \((a+b)^n\) is given by the formula \(T_k = \binom{n}{k} a^{n-k} b^k\). For the expression \((x + x^{-1})^8\), let \(a = x\) and \(b = x^{-1}\). Then, the general term is \(T_k = \binom{8}{k} x^{8-k} (x^{-1})^k\). Simplifying this, we get \(T_k = \binom{8}{k} x^{8 - 2k}\).
02
Find the Condition for the Constant Term
The expression inside involves \(x^{8-2k}\). For this term to be a constant, the power of \(x\) must be zero. Set the exponent equal to zero: \[8 - 2k = 0\].
03
Solve for the Value of k
Solve the equation \(8 - 2k = 0\) to find \(k\):Add \(2k\) to both sides to get \(8 = 2k\).Divide both sides by 2, resulting in \(k = 4\).
04
Substitute k back into the General Term
Substitute \(k = 4\) back into the expression for \(T_k\):\[T_4 = \binom{8}{4} x^{8 - 2(4)} = \binom{8}{4} x^{0}\]This simplifies to \(T_4 = \binom{8}{4} = 70\), since \(x^0 = 1\).
05
State the Constant Term
The constant term in the expansion is given by the coefficient of \(x^0\), which we found as \(70\). Therefore, the constant term in the expansion is 70.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial expansion
The Binomial Theorem is a powerful tool used to expand expressions raised to a power. It applies to any binomial expression of the form \((a + b)^n\).
This theorem helps us express the power as a sum of terms involving coefficients, powers of \(a\), and powers of \(b\).
Here鈥檚 how it looks in action:
This theorem helps us express the power as a sum of terms involving coefficients, powers of \(a\), and powers of \(b\).
Here鈥檚 how it looks in action:
- For expression \((a + b)^n\), each term in the expansion consists of a coefficient, a power of \(a\), and a power of \(b\).
- The coefficients are determined by combinations, which we will delve into later.
- The exponents of \(a\) decrease from \(n\) to 0, while the exponents of \(b\) increase from 0 to \(n\).
Exponent rules
Understanding exponent rules is crucial for simplifying expressions like \((x + x^{-1})^8\). Exponents let us say how many times to use a number in a multiplication. Here are some simple rules that come handy:
- Product of Powers Rule: When multiplying two powers with the same base, add the exponents: \(x^a \cdot x^b = x^{a+b}\).
- Power of a Power Rule: When raising a power to another power, multiply the exponents: \((x^a)^b = x^{a\cdot b}\).
- Negative Exponent Rule: A negative exponent means to take the reciprocal: \(x^{-a} = \frac{1}{x^a}\).
- Zero Exponent Rule: Any number raised to the power of zero is 1: \(x^0 = 1\).
Combinatorics
Combinatorics is the branch of mathematics dealing with combinations of objects. It is vital for understanding coefficients in a binomial expansion. The coefficient of each term is determined by the "choose" function, represented as \(\binom{n}{k}\).
- \(\binom{n}{k}\) is read as "n choose k" and counts how many ways \(k\) items can be chosen from \(n\) items without regard to order.
- It is calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(!\) denotes factorial, the product of all positive integers up to that number.
General term of a binomial expansion
The general term offers a concise way to determine any term within a binomial expansion. For \((a + b)^n\), the general term is given by \[T_k = \binom{n}{k} a^{n-k} b^k\].
This formula allows us to choose any term by simply plugging in values:
This formula allows us to choose any term by simply plugging in values:
- n is the power to which the binomial is raised.
- k is the current term position (starting from zero).
