Problem 34 - Algebra and Trigonometry with Analytic Geometry

Question

Without expanding completely, find the indicated term(s) in the expansion of the expression. \(\left(s-2 t^{3}\right)^{12}\) last three terms

Step-by-Step Solution

Answer

The last three terms are: \( 67584 s^2 t^{30}, -24576 s t^{33}, 4096 t^{36} \).

Step 1: Understand the Problem

We need to find the last three terms in the expansion of \( (s - 2t^3)^{12} \) without expanding the entire expression. This involves using the Binomial Theorem, which provides a formula to find specific terms in the expansion of a binomial expression.

Step 2: Apply the Binomial Theorem

The Binomial Theorem states that \( (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \). Here, \(n=12\), \(x = s\), and \(y = -2t^3\). We are interested in the last three terms, which correspond to \(k=10, 11, 12\).

Step 3: Find the Term for k=10

For \(k=10\), the term is computed as follows:\[ T_{10} = \binom{12}{10} s^{12-10} (-2t^3)^{10} = \binom{12}{10} s^2 (-2t^3)^{10} \]\[ T_{10} = \binom{12}{10} s^2 (-1)^{10} 2^{10} (t^3)^{10} \]\[ T_{10} = 66 s^2 (1024) t^{30} = 67584 s^2 t^{30} \]

Step 4: Find the Term for k=11

For \(k=11\), the term is calculated as:\[ T_{11} = \binom{12}{11} s^{12-11} (-2t^3)^{11} = \binom{12}{11} s (-2t^3)^{11} \]\[ T_{11} = 12 s (-1)^{11} 2^{11} (t^3)^{11} \]\[ T_{11} = 12 s (-2048) t^{33} = -24576 s t^{33} \]

Step 5: Find the Term for k=12

For \(k=12\), the term is:\[ T_{12} = \binom{12}{12} s^{12-12} (-2t^3)^{12} = (-2t^3)^{12} \]\[ T_{12} = (-1)^{12} 2^{12} (t^3)^{12} \]\[ T_{12} = 4096 t^{36} \]

Step 6: Write the Final Three Terms

Now combine the terms from \(k=10, 11, 12\):1. \( 67584 s^2 t^{30} \)2. \( -24576 s t^{33} \)3. \( 4096 t^{36} \)Thus, the last three terms of the expansion are: \( 67584 s^2 t^{30}, -24576 s t^{33}, 4096 t^{36} \).

Key Concepts

Binomial Expansion

When dealing with binomials, understanding how to expand them is crucial. A binomial consists of two terms. In our exercise, the binomial is \( (s-2t^3) \). The binomial expansion formula helps us expand expressions of the form \( (x+y)^n \). By expanding, we calculate each term in this expression when raised to a power \( n \).
To avoid expanding the entire expression, we use the Binomial Theorem. It guides us to find specific terms within the expansion. For instance, if you only need the last three terms, you can focus on those without calculating every preceding term.

• The Binomial Theorem formula is:\[(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k\]
• In our problem, we set \( x = s \), \( y = -2t^3 \), and \( n = 12 \).
• This formula helps us calculate the coefficients and powers of each term.

By applying the theorem strategically, we can find the required terms efficiently.

Combinatorial Coefficients

An essential part of the binomial expansion process is determining the right coefficients for each term. These coefficients are known as combinatorial coefficients and can be found using combinations, represented by \( \binom{n}{k} \). The role of these coefficients is to ensure each term is the correct size in relation to its position in the expansion.
Combinatorial coefficients are calculated as follows:

• The formula for \( \binom{n}{k} \) is \( \frac{n!}{k!(n-k)!} \), where \( n! \) is the factorial of \( n \).
• For our exercise, with \( n = 12 \), calculating combinatorial coefficients for \( k = 10, 11, \text{ and } 12 \) is key to finding the last three terms.
• For example, \( \binom{12}{10} \) results in a coefficient of 66, and \( \binom{12}{11} \) results in 12, showing how different stages in the process depend on these coefficients.

By mastering how to calculate these coefficients, you can solve binomial expansions much more easily.

Algebraic Expressions

Algebraic expressions can take on various forms, often composed of variables, numbers, and operations. These expressions are the building blocks for more complex operations. In binomial expansion, each term in an expanded expression is an algebraic expression.
Each term's simplicity in expression belies the complexity of the entire equation. With different powers of variables, expressions tell us about quantity and relation. Understanding them is critical when dealing with expanded forms.

• In our example, terms like \( 67584 s^2 t^{30} \) and \( -24576 s t^{33} \) are algebraic expressions resulting from expansion.
• The organization of terms—where the powers decrease for \( s \) and increase for \( t \)—displays the systematic nature of binomial expansion.
• When dealing with negative coefficients, as in the term \( -24576 st^{33} \), the minus sign indicates a subtraction in the polynomial setting.

Understanding these expressions' mechanics makes interpreting and using results from binomial expansions much simpler.