Problem 31 - Algebra and Trigonometry with Analytic Geometry

Question

Without expanding completely, find the indicated term(s) in the expansion of the expression. $$ \left(3 c^{2 / 5}+c^{4 / 5}\right)^{25} ; \quad \text { first three terms } $$

Step-by-Step Solution

Answer

First three terms: \(3^{25}c^{10}\), \(25 \cdot 3^{24}c^{52/5}\), \(300 \cdot 3^{23}c^{54/5}\).

Step 1: Understand the Use of Binomial Theorem

We need to apply the Binomial Theorem to find specific terms in the expansion. The Binomial Theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k}a^{n-k}b^k\). In this case, we identify \(a = 3c^{2/5}\) and \(b = c^{4/5}\) with \(n = 25\).

Step 2: Find the First Term

The first term corresponds to \(k=0\) in the binomial expansion. We use the formula: \(T_0 = \binom{25}{0}(3c^{2/5})^{25}(c^{4/5})^0\). Calculating each part: \(\binom{25}{0} = 1\), \((3c^{2/5})^{25} = 3^{25}c^{10}\), and \((c^{4/5})^0 = 1\). Therefore, the first term is \(3^{25}c^{10}\).

Step 3: Find the Second Term

The second term corresponds to \(k=1\). We use the formula: \(T_1 = \binom{25}{1}(3c^{2/5})^{24}(c^{4/5})^1\). Calculating: \(\binom{25}{1} = 25\), \((3c^{2/5})^{24} = 3^{24}c^{48/5}\), and \((c^{4/5})^1 = c^{4/5}\). Combine them to get the second term: \(25 \cdot 3^{24}c^{52/5}\).

Step 4: Find the Third Term

The third term corresponds to \(k=2\). The formula is: \(T_2 = \binom{25}{2}(3c^{2/5})^{23}(c^{4/5})^2\). Calculating: \(\binom{25}{2} = 300\), \((3c^{2/5})^{23} = 3^{23}c^{46/5}\), and \((c^{4/5})^2 = c^{8/5}\). Combine them to find the third term: \(300 \cdot 3^{23}c^{54/5}\).

Key Concepts

Algebra

Algebra forms a crucial part of solving problems involving expressions and equations. It deals with symbols and the rules for manipulating these symbols to solve equations.
In the context of polynomial expansion, algebraic techniques allow us to re-arrange and combine terms in mathematical expressions.

• Algebra provides a framework to handle expressions systematically.
• It introduces techniques for simplifying expressions and finding values of unknown variables.
• By understanding algebra, we can apply formulas like the Binomial Theorem efficiently.

In our exercise, using algebra helps transform and simplify the terms resulting from the binomial expansion. By following the rules of algebra, terms from the expansion are systematically calculated without expanding the whole expression.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. They are crucial in the process of expanding polynomials or any expressions involving powers.

• Exponents follow specific rules such as product of powers, power of a power, and power of a product.
• Knowing these rules assists us in simplifying complex algebraic expressions.

In our exercise, exponents are used extensively to determine the power to which each term in the polynomial expansion is raised.
For instance, when we see \(3c^{2/5}\), it means 3 is multiplied by \(c^{2/5}\) a certain number of times depending on its exponent in the expansion. By consistently applying exponent rules, we ensure that each term's calculations are accurate and coherent.

Polynomial Expansion

Polynomial expansion involves expressing a power of a binomial (two-term expression) in expanded form. The Binomial Theorem is the primary tool for this task, enabling the systematic expansion of binomials raised to any integer power.

• Each term in the expansion is calculated using a specific formula which involves coefficients, powers of the first term, and powers of the second term.
• These terms are then summed up to form the expanded polynomial.

In our given exercise, polynomial expansion is applied using the Binomial Theorem to find specific terms instead of calculating the entire expanded polynomial. This targeted approach saves time and simplifies the calculations.
By finding just the terms needed for \((3c^{2/5} + c^{4/5})^{25}\), we efficiently solve the problem without unnecessary computation.

Combinations

Combinations are an essential aspect of the Binomial Theorem, representing the coefficients of terms in a polynomial expansion. The combination formula, denoted as \(\binom{n}{k}\), is used to determine how many ways \(k\) elements can be chosen from \(n\) elements, without considering the order.
This concept is crucial because it tells us the coefficient for each term in the polynomial expansion.

• The calculation of \(\binom{n}{k}\) provides the number of combinations of terms taken \(k\) at a time.
• Each term's coefficient in the binomial expansion is determined using this formula.

In solving our exercise, these combinations help us compute terms like \(\binom{25}{0}\), \(\binom{25}{1}\), and \(\binom{25}{2}\), which are integral to calculating the specific terms in the expansion. Understanding how to calculate combinations is fundamental to using and applying the Binomial Theorem correctly.