Problem 29 - Algebra and Trigonometry

Question

Find the middle term in the expansion of \(\left(x^{2}+1\right)^{18}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Total Number of Terms

The expansion of \(a+b\)^n has \(n+1\) terms. Here, the expression is \(\left(x^2+1\right)^{18}\), so there are \(18+1=19\) terms.

Step 2: Determine the Middle Term Position

Since there are 19 terms, the middle term is the 10th term in the sequence of terms.

Step 3: Use the Binomial Theorem Formula

The \(r+1\)th term in the expansion of \(\left(a+b\right)^n\) is given by \(T_{r+1} = \binom{n}{r} a^{n-r} b^r\). For the middle term, \(r=9\), \(a=x^2\), and \(b=1\).

Step 4: Calculate the 10th Term

Plugging \(n=18\), \(a=x^2\), and \(b=1\) into \(T_{10}=\binom{18}{9} (x^2)^{18-9} (1)^9\) gives \(T_{10}=\binom{18}{9} (x^2)^9\).

Step 5: Simplify the Middle Term

Since \(b=1\), the term simplifies to \(T_{10}=\binom{18}{9} x^{18}\). Calculating \(\binom{18}{9} = 48620\), we find \(T_{10}=48620 x^{18}\).

Key Concepts

Middle Term

When you perform a binomial expansion, it typically results in a sequence of terms. In this sequence, the middle term is essential as it is often the most significant or highlighted part of the expansion. To find the middle term, you first need to determine the total number of terms from the expression \(\left(a+b\right)^n\). This expression yields \(n+1\) terms.

For instance, in the expansion of \(\left(x^2+1\right)^{18}\), we calculate \(18+1=19\) terms. Hence, the middle term will be the 10th term, given an odd number of terms. Identifying the middle term can help in estimating the approximate values for large binomial expansions or simplifying a larger problem into smaller parts.

Binomial Expansion

The binomial expansion allows you to express a power of a binomial as a sum of terms. It follows a specific theorem, commonly known as the Binomial Theorem. This theorem provides a formula to describe the expansion of \(\left(a+b\right)^n\) as a series of binomial coefficients multiplied by powers of the individual terms.

The formula is given by: \[ \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r \] Here, \(\binom{n}{r}\) is the binomial coefficient, which counts the number of combinations of \(n\) items taken \(r\) at a time. Each term in the expansion represents a combination of different powers of \(a\) and \(b\) that add up to \(n\).

• This pattern continues until all terms are formed in the expansion.
• It can simplify computations and provide insight into patterns in algebraic expressions.

Combinatorics

Combinatorics is the branch of mathematics focused on counting, arranging, and finding patterns. In binomial expansions, combinatorics plays a critical role through binomial coefficients.

These coefficients \(\binom{n}{r}\) indicate the number of ways to choose \(r\) elements from \(n\) without regard to order. This concept is crucial to configuration tasks, optimization problems, and algebra.

• For the polynomial \(\left(x^2+1\right)^{18}\), the middle term coefficient is calculated using combinatorics: \(\binom{18}{9}\).
• Such combinations help determine coefficients for diverse portions of algebraic expressions.
• Combinatorial methods simplify complex problems into solvable combinations.

Understanding these elements gives you a foundation for comprehending more intricate subjects in mathematics.

Polynomial Expansion

Polynomial expansion refers to the process of expressing a polynomial in a longer, expanded form. In the context of the binomial expansion, this involves breaking down a binomial raised to a power into a series of terms. These terms are derived by multiplying different powers of the variables within the binomial.

Consider the polynomial expansion of \(\left(x^2+1\right)^{18}\); this process results in a sequence of 19 terms. Each term consists of an integral power of \(x^2\) combined with a constant factor.

• These constant factors are binomial coefficients, calculated via combinatorial principles.
• The expansion of a polynomial helps you understand its behavior and how each part contributes to the overall expression.

By expanding polynomials, you gain insights into the relationships among terms and their powers.