Problem 7 - Discrete Mathematics with Applications

Question

Using the binomial theorem, expand each. $$(2 x-1)^{5}$$

Step-by-Step Solution

Answer

Using the binomial theorem, we can find the expanded form of \((2x-1)^5\) as follows: \((2x-1)^5 = 32x^5 - 80x^4 + 80x^3 - 40x^2 + 10x - 1\).

Step 1: Identify the values of a, b, and n

In the expression \((2x-1)^5\), we can notice that \(a = 2x\), \(b = -1\) and \(n = 5\). We will use these values in the binomial theorem formula.

Step 2: Apply the binomial theorem

Using the binomial theorem, we can rewrite the expression as a sum of the form: $$ (2x - 1)^5 = \sum_{k=0}^{5} \binom{5}{k} (2x)^{5-k} (-1)^{k} $$

Step 3: Calculate the sum

Now, we will find the expanded form by calculating each term of the sum: $$ (2x - 1)^5 = \binom{5}{0} (2x)^5(-1)^0 + \binom{5}{1} (2x)^4(-1)^1 + \binom{5}{2} (2x)^3(-1)^2 + \binom{5}{3} (2x)^2(-1)^3 + \binom{5}{4} (2x)^1(-1)^4 + \binom{5}{5} (2x)^0(-1)^5 $$ Compute the binomial coefficients and value of each term: $$ (2x-1)^5 = 1 \cdot 32 x^5 \cdot 1 - 5 \cdot 16 x^4 \cdot 1 + 10 \cdot 8 x^3 \cdot 1 - 10 \cdot 4 x^2 \cdot 1 + 5 \cdot 2 x \cdot 1 - 1 \cdot 1 \cdot 1 $$ Simplify each term and combine: $$ (2x-1)^5 = 32x^5 - 80x^4 + 80x^3 - 40x^2 + 10x - 1 $$ The expanded form of the given expression, \((2x-1)^5\), is \(32x^5 - 80x^4 + 80x^3 - 40x^2 + 10x - 1\).

Key Concepts

Binomial Coefficients

At the heart of the binomial theorem lies a fundamental mathematical concept known as binomial coefficients. These coefficients are the numbers that appear in the expanded form of a binomial expression, typically represented in combinatorial mathematics as \( \binom{n}{k} \), where 'n' is the exponent of the binomial and 'k' ranges from 0 to n. A clearer definition of binomial coefficients is that they represent the number of ways to choose 'k' elements from a set of 'n' elements without regard to order.

For instance, in the exercise \((2x-1)^5\), we calculate the coefficients \( \binom{5}{k} \) for each term of the expansion where 'k' ranges from 0 to 5. They correspond to 1, 5, 10, 10, 5, and 1, respectively for \( \binom{5}{0} \), \( \binom{5}{1} \), \( \binom{5}{2} \), \( \binom{5}{3} \), \( \binom{5}{4} \), and \( \binom{5}{5} \). These coefficients are central to determining the multiplied factor for each corresponding term of \((2x)^{5-k}(-1)^k\) in the polynomial expansion.

Polynomial Expansion

Polynomial expansion, in algebra, refers to the process of expressing a polynomial that has been raised to a power as a sum of terms consisting of coefficients multiplied by the variables to various powers. For binomial expressions like \((a+b)^n\), the binomial theorem provides a systematic way to perform these expansions. It finds use in multiple areas of mathematics and applied science fields where simplification of algebraic expressions is crucial.

When we apply the binomial theorem to expand \((2x-1)^5\), we are in fact converting the binomial into a polynomial with six terms. Each term in the sum is the product of a binomial coefficient, a power of \(2x\), and a power of \(-1\). This polynomial is already in its expanded form, and it is a concrete example of how the binomial theorem aids in the simplification of complex algebraic expressions that otherwise might be tedious to work out manually.

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and arithmetic operations. Expressions can be as simple as \(x + 3\) or as complex as \((2x-1)^5\), as shown in our exercise. Such expressions become powerful tools in formulating and solving equations and represent relationships between variables and constants.

Algebraic expressions are the building blocks of algebra and allow for the generalization of mathematical ideas. The importance of understanding how to manipulate these expressions can't be overstated for any student of mathematics. In our example, \((2x-1)^5\) is an algebraic expression that involves a binomial raised to the fifth power. By expanding this expression using the binomial theorem, we transform it into a polynomial, which is a specific type of algebraic expression that has a sum of monomials, each consisting of a coefficient and a variable raised to a non-negative integer power.