Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Without expanding completely, find the indicated term(s) in the expansion of the expression. \(\left(r s^{2}+t\right)^{7}\) two middle terms

Step-by-Step Solution

Answer

The middle terms are \(35 r^4 s^8 t^3\) and \(35 r^3 s^6 t^4\).

Step 1: Identify the Expression

The expression given is \((rs^2 + t)^7\). We need to find the two middle terms in its expansion.

Step 2: General Term in Binomial Expansion

The general term \(T_k\) in the expansion of \((rs^2 + t)^n\) is given by the formula: \[ T_k = \binom{n}{k} (rs^2)^{n-k} t^k \]Here, \(n = 7\).

Step 3: Determine Number of Terms

The number of terms in the expansion of a binomial expression \((a + b)^n\) is \(n + 1\). Therefore, there are 8 terms in total for \((rs^2 + t)^7\).

Step 4: Identify Middle Terms

Since there are 8 terms, the two middle terms are the 4th and 5th terms. These are given by the terms where \(k = 3\) and \(k = 4\).

Step 5: Calculate the 4th Term (k = 3)

Using the general term formula:\[ T_3 = \binom{7}{3} (rs^2)^{7-3} t^3 = \binom{7}{3} (rs^2)^4 t^3 \]Simplify further after calculating the binomial coefficient and powers.

Step 6: Calculate the 5th Term (k = 4)

Using the general term formula:\[ T_4 = \binom{7}{4} (rs^2)^{7-4} t^4 = \binom{7}{4} (rs^2)^3 t^4 \]Simplify further after calculating the binomial coefficient and powers.

Step 7: Final Result

Calculate the binomial coefficients \(\binom{7}{3} = 35\) and \(\binom{7}{4} = 35\).Thus, the two middle terms are:\[ T_3 = 35 (rs^2)^4 t^3 = 35 r^4 s^8 t^3 \]\[ T_4 = 35 (rs^2)^3 t^4 = 35 r^3 s^6 t^4 \]

Key Concepts

General Term in Binomial Expansion

In mathematics, binomial expansion is a powerful tool that helps us express an expression raised to a power without having to manually multiply it out. The general term in the expansion of a binomial expression \( (a + b)^n \) is formulated as follows:

• \( T_k = \binom{n}{k} a^{n-k} b^k \)

This formula allows us to easily determine any specific term in the expansion. Here,

• \( T_k \) is the \( k^{th} \) term,
• \( \binom{n}{k} \) is the binomial coefficient,
• \( a \) and \( b \) are the terms being adjusted in each binomial expression,
• and \( n \) is the power to which the binomial expression is raised.

By using this formula, you don't need to expand the expression fully. Instead, you can directly jump to the term you're interested in. For example, if you're given an expression like

• \( (rs^2 + t)^7 \).

You can plug parameters into the formula to get specific terms in the expansion.

Binomial Coefficient

The binomial coefficient is a critical part of the binomial expansion, and it's denoted by \( \binom{n}{k} \). This coefficient essentially finds out how many different ways you can choose \( k \) items from a set of \( n \) items, which is also why it's often called a "combinatorial coefficient."

• The formula for calculating it is: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), where \("!"\) signifies factorial, meaning the product of all positive integers up to that number.

In the context of binomial expansion, the binomial coefficient gives the weight of each term in the expanded series. For instance, in the expression \( (rs^2 + t)^7 \), you would use \( \binom{7}{k} \) to find the coefficient for various terms. This enables you to know precisely how prominent or subtle each term's contribution is to the whole expansion. When you master the binomial coefficient, it becomes a lot easier to compute the desired terms in complex expressions.

Middle Terms of Expansion

Identifying middle terms in a binomial expansion can be especially useful when the expanded series is lengthy. In any binomial expansion such as \((a + b)^n\), there are \( n+1 \) total terms. Therefore:

• If \( n+1 \) is even, the two middle terms lie exactly at positions \( \frac{n+1}{2} \) and \( \frac{n+1}{2} + 1\).
• If \( n+1 \) is odd, there's a single middle term at \( \frac{n+1}{2} \).

For example,in the expression \((rs^2 + t)^7\), there are total 8 terms \((7 + 1)\). These 8 terms mean our middle terms are precisely the 4th and 5th terms. So, to find these terms, substitute \( k = 3 \) and \( k = 4 \) in the general term formula.

• The calculations have shown that these middle terms simplify to: \\( T_3 = 35 r^4 s^8 t^3 \) and \\( T_4 = 35 r^3 s^6 t^4 \).

By understanding how to spot and calculate the middle terms, you can narrow down on the most balanced terms in the expansion without needing to list out the entire expanded form.