Question

Write the requested term of each binomial expansion, and simplify. Fifth term of \((x-2 \sqrt{y})^{25}\)

Short answer

The fifth term of the binomial expansion \((x-2 \sqrt{y})^{25}\) is \(202,400x^{21}y^2\).

Step-by-step solution

Identifying the General Term in Binomial Expansion

In a binomial expansion of \((a+b)^n\), the k-th term is given by \(T_k = \binom{n}{k-1} a^{n-(k-1)} b^{k-1}\), where \(\binom{n}{k-1}\) is the binomial coefficient. In this case we want the fifth term, so we set \(k = 5\).

Substitute the values into the Binomial Theorem

For the binomial \((x - 2 \sqrt{y})^{25}\), the a is \(x\) and the b is \(-2 \sqrt{y}\). Plugging in the values we get \(T_5 = \binom{25}{5-1} x^{25-(5-1)} (-2 \sqrt{y})^{5-1}\).

Calculating the Binomial Coefficient

Calculate \(\binom{25}{4}\) which is the coefficient of the fifth term. \(\binom{25}{4} = \frac{25!}{4!(25-4)!}\).

Simplifying the Exponents

Simplify the exponents of x and \(-2 \sqrt{y}\) to get \(T_5 = \binom{25}{4} x^{21} (-2 \sqrt{y})^4\).

Solving Binomial Coefficient

The binomial coefficient \(\binom{25}{4}\) equals \(\frac{25 \times 24 \times 23 \times 22}{4 \times 3 \times 2 \times 1} = 12,650\).

Simplifying the Term

Raise \(-2 \sqrt{y}\) to the fourth power, which results in \((-2)^4 (\sqrt{y})^4 = 16y^2\). Now multiply this by the binomial coefficient and \(x^{21}\) to get the simplified fifth term: \(T_5 = 12,650 \times x^{21} \times 16y^2\).

Combine the Coefficients

Multiply the coefficients to find the simplified term. \(T_5 = 12,650 \times 16 \times x^{21} \times y^2\).

Final Simplification

The final calculation results in \(T_5 = 202,400 \times x^{21} \times y^2\).

Key concepts

Binomial Theorem

The binomial theorem is a fundamental mathematical principle that describes the expansion of powers of a binomial, an expression involving the sum or difference of two terms. For instance, when we have an algebraic expression like \( (a + b)^n \), where \( n \) is a non-negative integer, the binomial theorem allows us to expand this expression into a sum involving terms of the form \( a^k b^{n-k} \), multiplied by a corresponding binomial coefficient.

The formula for the expansion is given by: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] This means that the expanded form will have \( n+1 \) terms, where each term is a product of a binomial coefficient, a power of \( a \), and a power of \( b \).

This principle is used widely to solve various types of problems in algebra, calculus, probability, and more. It's particularly handy in situations where raising a binomial to a power by standard multiplication would be impractical due to a high exponent.

Binomial Coefficient

A binomial coefficient, denoted as \( \binom{n}{k} \), represents the number of ways to choose \( k \) elements out of a set of \( n \) distinct elements without regard to the order of selection. It is also known as a 'combination' in combinatorics.

The binomial coefficient is an integral part of the binomial theorem. It's calculated using the formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] where \( ! \) denotes factorial, the product of all positive integers up to that number. For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 \).

These coefficients appear in the expanded form of a binomial expression, determining the weight of each term in the expansion. Understanding how to compute and apply these coefficients is crucial when working with binomial expansions, as they greatly simplify the process of determining the individual terms.

Algebraic Expression Simplification

Simplifying algebraic expressions is a process of rewriting them in their most basic form without changing their value. This involves combining like terms, reducing fractions, and factoring, among other operations. It is an essential skill for solving equations and understanding various areas of mathematics.

In the context of binomial expansion, simplification might involve calculating the value of binomial coefficients, raising terms to their respective powers, and then multiplying everything together to get a compact, simplified expression.

For example, in the case of \( x^{21}(-2 \sqrt{y})^4 \), simplification would involve squaring \( \sqrt{y} \) to get \( y \) and raising -2 to the fourth power, which would be 16, and then multiplying this by the previously calculated binomial coefficient. The end goal is to present the expanded term in its simplest, most digestible form. Simplification helps in understanding and working with the algebraic expressions more effectively and is often the concluding step in mathematical problem-solving.