Problem 46 - A Graphical Approach to College Algebra

Question

Find the two middle terms of \(\left(-2 m^{-1}+3 n^{-2}\right)^{11}\).

Step-by-Step Solution

Answer

The two middle terms are \(7185024 \, m^{-6} n^{-10}\) and \(-10777536 \, m^{-5} n^{-12}\).

Step 1: Identify the middle terms

For the binomial expansion of \((-2m^{-1} + 3n^{-2})^{11}\), since \(n = 11\) (odd), there are \(n+1 = 12\) terms. The two middle terms are the 6th and 7th terms (i.e., \(T_6\) and \(T_7\), corresponding to \(k = 5\) and \(k = 6\) in the binomial formula).

Step 2: Compute the 6th term (k = 5)

\(T_6 = \binom{11}{5}(-2m^{-1})^{11-5}(3n^{-2})^5 = \binom{11}{5}(-2)^6 m^{-6} \cdot 3^5 n^{-10}\)
\(= 462 \cdot 64 \cdot 243 \cdot m^{-6} n^{-10}\)
\(= 7185024 \, m^{-6} n^{-10}\)

Step 3: Compute the 7th term (k = 6)

\(T_7 = \binom{11}{6}(-2m^{-1})^{11-6}(3n^{-2})^6 = \binom{11}{6}(-2)^5 m^{-5} \cdot 3^6 n^{-12}\)
\(= 462 \cdot (-32) \cdot 729 \cdot m^{-5} n^{-12}\)
\(= -10777536 \, m^{-5} n^{-12}\)

Step 4: Final answer

The two middle terms are:
\(T_6 = 7185024 \, m^{-6} n^{-10}\)
\(T_7 = -10777536 \, m^{-5} n^{-12}\)

Key Concepts

Understanding Binomial Expansion

The Binomial Theorem is a powerful tool that allows us to expand expressions raised to a power, like \((a+b)^n\). It's based on a simple idea that involves breaking down a binomial (two terms) into a sum of terms involving powers of both terms. This expansion involves binomial coefficients and various powers of the two terms.
For example, in the expression \((-2 m^{-1} + 3 n^{-2})^{11}\), we can expand it into a sum of terms using the formula: \[ (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] Here, \(a = -2 m^{-1}\) and \(b = 3 n^{-2}\), and we're expanding for \(n = 11\). This formula essentially allows us to find every term of the expanded version of the expression. Each term in the expansion takes the form of these alternating powers of \(a\) and \(b\) combined with binomial coefficients.

Finding Specific Terms in a Binomial Expansion

Finding specific terms in a binomial expansion is about using the formula to isolate the terms that interest us. For our expression \((-2 m^{-1} + 3 n^{-2})^{11}\), the 'middle' terms are important.
In an expansion of \(n+1\) terms, if \(n\) is odd, like 11, there will be two middle terms at positions \(\frac{n}{2}\) and \(\frac{n}{2} + 1\), which would be terms at \(k=5\) and \(k=6\).Each of these terms is calculated using the formula: \[ \text{Term}_{k+1} = \binom{n}{k} \cdot a^{n-k} \cdot b^k \] To find the terms at \(k=5\) and \(k=6\), you substitute back into the formula, calculating:

• The binomial coefficient \(\binom{11}{5}\) and \(\binom{11}{6}\).
• The powers \(a^{11-5} = (-2 m^{-1})^6\) and \(b^{5} = (3 n^{-2})^5\).
• The powers \(a^{11-6} = (-2 m^{-1})^5\) and \(b^{6} = (3 n^{-2})^6\).

The Role of Binomial Coefficients

Binomial coefficients are key to understanding how terms in a binomial expansion are constructed. They are written as \(\binom{n}{k}\), which is read as 'n choose k', and calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \(n!\) (n factorial) means the product of all positive integers up to \(n\).In our specific problem, we needed to calculate \(\binom{11}{5}\) and \(\binom{11}{6}\). These coefficients determine the weight of each term in the expansion.
They serve as multipliers for the combinations of \(a\) and \(b\) terms to ensure each arrangement is considered correctly. Thus, they play a critical role in both finding the specific terms of interest and in the overall structure of the polynomial's expansion.