Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(x^{3}+5 x^{-2}\right)^{20}, \quad \text { first three terms } $$

Step-by-Step Solution

Answer

The first three terms are \(x^{60}, 100x^{55}, \text{and } 4750x^{50}\).

Step 1: Understand the Binomial Expansion Formula

The expression \((x^3 + 5x^{-2})^{20}\) can be expanded using the Binomial Theorem, which states that \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]where \(\binom{n}{k}\) is the binomial coefficient. We will use this formula to identify the first three terms in the expansion.

Step 2: Identify Terms Using Binomial Coefficients

For the expression \((x^3 + 5x^{-2})^{20}\), identify the first three terms as follows:- **First Term:** For \(k = 0\), term is \(\binom{20}{0} (x^3)^{20} (5x^{-2})^0 = x^{60}\).- **Second Term:** For \(k = 1\), term is \(\binom{20}{1} (x^3)^{19} (5x^{-2})^1 = 20 \cdot x^{57} \cdot 5x^{-2} = 100x^{55}\).- **Third Term:** For \(k = 2\), term is \(\binom{20}{2} (x^3)^{18} (5x^{-2})^2 = 190 \cdot x^{54} \cdot 25x^{-4} = 4750x^{50}\).

Key Concepts

Binomial Coefficients

Understanding binomial coefficients is crucial when dealing with expressions like \((x^3 + 5x^{-2})^{20}\). These coefficients, denoted as \(\binom{n}{k}\), represent the number of ways to choose \(k\) items from \(n\) items without regard to order. They are key to determining the specific terms in a binomial expansion.

In the expression \((x^3 + 5x^{-2})^{20}\), the binomial coefficients help us figure out the weight of each term. For example:
- For \(k=0\), the binomial coefficient is \(\binom{20}{0} = 1\).
- For \(k=1\), the binomial coefficient is \(\binom{20}{1} = 20\).
- For \(k=2\), the binomial coefficient is \(\binom{20}{2} = 190\).

These coefficients multiply the powers of each term, allowing us to determine the overall contribution each part of the original expression makes to the final expanded polynomial.

Polynomial Expansion

The concept of polynomial expansion involves expressing a power of a binomial expression as a sum of terms. Thanks to the Binomial Theorem, we can expand expressions like \((x^3 + 5x^{-2})^{20}\) without multiplying everything manually, term by term.

In our expression, the expansion takes the form of a series:

• Each term involves a combination of powers of both \(x^3\) and \(5x^{-2}\).
• The powers of \(x\) and the constant multipliers are determined systematically by the binomial coefficients.

For instance, in the expansion, the first few terms are:

• The first term is when \(k=0\): \(x^{60}\).
• The second term is when \(k=1\): \(100x^{55}\).
• The third term is when \(k=2\): \(4750x^{50}\).

Each term follows the pattern of combining the coefficients with the corresponding powers, leading to a compressed form that showcases the expansion.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In our exercise, we started with \((x^3 + 5x^{-2})^{20}\), which is a binomial expression raised to a power.

Expressions like \(x^3\) and \(5x^{-2}\) highlight the utility of algebraic symbols, allowing us to compress complex mathematical ideas into a more manageable form. When these are raised to a power, like 20 in our case, they introduce a layer of complexity that requires expansion methods like the binomial theorem to simplify.

Moreover, each individual term in an algebraic expression can be further described by its parts:

• Coefficients: The constant multipliers in front of variables, such as 5 in \(5x^{-2}\).
• Variables: Symbols like \(x\) that represent numbers.
• Exponents: Indicate the power to which a variable is raised, such as \(3\) in \(x^3\).

This transformation from a compact exponential form to an expanded polynomial expression showcases the power and flexibility of algebra in breaking down complex expressions.