Problem 32 - Precalculus with Limits

Question

In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(2x - 5y\right)^5 \)

Step-by-Step Solution

Answer

The expanded form of \( \left(2x - 5y\right)^5 \) is \(32x^5 - 800x^4y + 8000x^3y^2 - 40000x^2y^3 + 100000xy^4 - 31250y^5\)

Step 1: Identify terms

First identify the two terms of the binomial. For this problem, the terms of the binomial are \(2x\) and \(-5y\). Note that we are considering \(-5y\) as one term, not two separate terms.

Step 2: Expand using the Binomial Theorem

The Binomial Theorem states for any integers \(n\) and \(r\), \((a+b)^n = \sum_{r=0}^{n} C(n, r) * a^{n-r} * b^r\), where C(n, r) are binomial coefficients, computed as \(C(n, r) = \frac{n!}{r!(n-r)!}\). Now, using the theorem to expand the expression: \( (2x - 5y)^5 = C(5, 0)*(2x)^5*(-5y)^0 + C(5, 1)*(2x)^4*(-5y)^1 + C(5, 2)*(2x)^3*(-5y)^2 + C(5, 3)*(2x)^2*(-5y)^3 + C(5, 4)*(2x)^1*(-5y)^4 + C(5, 5)*(2x)^0*(-5y)^5 \)

Step 3: Simplify the terms

Simplify each term in the expanded binomial. Remember that any number raised to the power of 0 is 1. This gets: \(32x^5 - 800x^4y + 8000x^3y^2 - 40000x^2y^3 + 100000xy^4 - 31250y^5\)

Step 4: Finalize the expansion

The final expanded form of \( (2x - 5y)^5 \) is : \(32x^5 - 800x^4y + 8000x^3y^2 - 40000x^2y^3 + 100000xy^4 - 31250y^5\)

Key Concepts

Understanding Binomial Expansion

Binomial expansion is an essential concept in algebra, used to expand expressions raised to a power, specifically combinations of two terms or binomials. It builds upon the Binomial Theorem, which provides a formula to express

• a binomial raised to an integer power
• in terms of sums of products of the terms of the binomial and binomial coefficients.

Take for instance, the expression \( (a + b)^n \), where "a" and "b" are the terms and "n" is the exponent. This can be expanded into a sum of terms involving powers of "a" and "b".
Each term in this expansion contributes its part according to a pattern based on the binomial coefficients. Each expansion is a finite series, with as many terms as the degree "n" plus one. Understanding this concept simplifies complex polynomial expressions, making it one of the foundational tools in algebra.

The Role of Factorials in Expansion

The concept of factorials plays a crucial part within the mechanism of calculating binomial coefficients. A factorial, represented by an exclamation point (e.g., \( n! \)), is the product of all positive integers up to a given number.
For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).
Factorials are specifically used in the formula for binomial coefficients:

• \( C(n, r) = \frac{n!}{r! (n - r)!} \)

where "n" is the power the binomial is raised to, and "r" indicates the term number in the expansion.
Factorials determine how coefficients are derived and directly affect the value of each expanded term. They make it possible to systematically calculate each coefficient needed for the binomial expansion, allowing for an efficient way to handle terms of higher powers.

Understanding Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients, involving only non-negative integer powers. The binomial expansion is a perfect practical application in deriving polynomial forms from binomials.
When a binomial like \((2x - 5y)^5\) is expanded, it transforms into a polynomial, where terms like \(32x^5\), \(-800x^4y\), etc., appear, reflecting different orders of variables in the expression. Each term consists of coefficients, which are numbers multiplied by the variables, and the degrees, which indicate the power the variable is raised to.
The result of a binomial expansion is a fully expanded polynomial expression that clearly represents the potential outcomes of distributing the binomial across its power, revealing all possibilities if you were to "expand" it manually.

Calculating Binomial Coefficients

Binomial coefficients are numbers that give the coefficients of the terms in the expansion of a binomial raised to a power. They are derived from Pascal’s Triangle and can be calculated using the formula for combinations:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

Binomial coefficients help determine the weight or importance of each term within the expansion, essentially stating how many ways a term can be arranged or "chosen." In the given problem of \((2x - 5y)^5\), the coefficients are calculated for each term in the expansion to determine its contribution to the final expression.
By organizing these coefficients systematically, binomial expansion not only simplifies polynomial creation but also enhances comprehension of mathematical relationships within algebra.