Problem 33 - Precalculus: Mathematics for Calculus

Question

Find the indicated terms in the expansion of the given binomial. The middle term in the expansion of \(\left(x^{2}+1\right)^{18}\).

Step-by-Step Solution

Answer

The middle term is \(48620 \cdot x^{18}\).

Step 1: Identify the Binomial Expansion Formula

The binomial expansion of \((a + b)^n\) is given by the formula \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). For this problem, \(a = x^2\), \(b = 1\), and \(n = 18\).

Step 2: Determine the Number of Terms

The expansion will contain \(n + 1\) terms, so the expansion of \((x^2 + 1)^{18}\) will contain 19 terms.

Step 3: Find the Middle Term Index

For an even \(n\), the middle term in the expansion is the \(\left( \frac{n}{2} + 1 \right)\)-th term. Here, \(n = 18\), so the middle term index is \(\frac{18}{2} + 1 = 10\).

Step 4: Use Binomial Coefficient to Find the Middle Term

The middle term (10th term) can be represented by \(T_{k+1}\) where \(k = 9\):\[T_{10} = \binom{18}{9} (x^2)^{18-9} (1)^9\]This simplifies to:\[T_{10} = \binom{18}{9} (x^2)^9\]

Step 5: Calculate the Binomial Coefficient

Compute \(\binom{18}{9}\):\[\binom{18}{9} = \frac{18 \times 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10}{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 48620\]

Step 6: Find the Expression of the Middle Term

Substitute the value of the binomial coefficient back into the expression:\[T_{10} = 48620 \cdot (x^2)^9\]Simplify:\[T_{10} = 48620 \cdot x^{18}\]

Key Concepts

Binomial Expansion

The binomial expansion is the method of expanding an expression that has been raised to any power, in the form \((a + b)^n\). This technique is derived from the Binomial Theorem, which was established by Isaac Newton. The theorem allows us to break down the expression into a sum of terms in a specific sequence, making it easier to handle polynomials raised to high powers.

• The basic idea is to separate the binomial into individual terms: \((a + b)^n = a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + \, \ldots \, + b^n\). Each term in the expansion is built by multiplying a power of the first term, \(a\), and a power of the second term, \(b\).
• These terms have coefficients known as "binomial coefficients," which we will explore further in the next section.
  
• To find a particular term in the expansion, you simply need to substitute the corresponding exponent values obtained from deciding which term of the expansion you are dealing with.
  

The binomial expansion allows for any polynomial to be expanded into a series of terms which are easier to manage and compute.

Binomial Coefficient

A key component of the binomial expansion is the binomial coefficient. The binomial coefficient is a numerical factor that multiplies each term in a binomial expansion. It indicates the number of ways to choose a subset of \(k\) elements, independent of order, from a set of \(n\) elements, commonly written as \(\binom{n}{k}\). You can calculate these coefficients using factorial notation:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• In the exercise, the binomial coefficient \(\binom{18}{9}\) had to be computed to find the middle term's coefficient, and was calculated to be 48620.

These coefficients are important because they tell us how many times a certain product of terms in the expansion appears when the binomial expression is expanded. Each term's coefficient in the expansion is dependent on these values, making them crucial for finding any particular term accurately.

Middle Term

When tackling binomial expansions, it's often useful to find specific terms in the sequence, such as the middle term. This is particularly true for large \(n\) values, where fully writing out the whole expansion isn't feasible. For even \(n\), the middle term is found by calculating \(\frac{n}{2} + 1\). This is straightforward once you understand that the expansion of \((x^2 + 1)^{18}\) contains \(n+1 = 19\) terms.

• To find this term, you determine the indices and plug them into the binomial expansion formula. For the example given, the 10th term corresponds to the middle term.
• By using the binomial coefficient, you pinpoint the exact power of the term and the overall coefficient multiplying your expression.

So in the given problem, the middle term's coefficient and power of \(x\) led to the term \(48620 \cdot x^{18}\), highlighting how you can use the binomial coefficient and formula to focus on individual components of a larger expansion.