Problem 33 - Beginning and Intermediate Algebra

Question

Use the binomial theorem to expand each expression. $$(w+2)^{4}$$

Step-by-Step Solution

Answer

Using the binomial theorem, the expanded form of the expression \((w + 2)^4\) is: \((w + 2)^4 = w^4 + 8w^3 + 24w^2 + 32w + 16\).

Step 1: Set up the binomial formula

Given the binomial theorem, we can rewrite our expression as follows: \((w + 2)^4 = \sum_{k=0}^{4} \binom{4}{k} w^{4-k} 2^k\)

Step 2: Calculate the binomial coefficients

Now calculate the binomial coefficients using the formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) Our coefficients are: \(\binom{4}{0} = \frac{4!}{0!4!} = 1\), \(\binom{4}{1} = \frac{4!}{1!3!} = 4\), \(\binom{4}{2} = \frac{4!}{2!2!} = 6\), \(\binom{4}{3} = \frac{4!}{3!1!} = 4\), and \(\binom{4}{4} = \frac{4!}{4!0!} = 1\)

Step 3: Substitute the coefficients into the formula

Substitute the calculated binomial coefficients and the given values of a and b into our initial expression: \[(w + 2)^4 = 1 \cdot w^{4-0} \cdot 2^0 + 4 \cdot w^{4-1} \cdot 2^1 + 6 \cdot w^{4-2} \cdot 2^2+ 4 \cdot w^{4-3} \cdot 2^3 + 1 \cdot w^{4-4} \cdot 2^4\]

Step 4: Simplify the expression

Simplify the expression by calculating the powers of w and 2: \[(w + 2)^4 = 1 \cdot w^4 \cdot 1 + 4 \cdot w^3 \cdot 2 + 6 \cdot w^2 \cdot 4 + 4 \cdot w^1 \cdot 8 + 1 \cdot w^0 \cdot 16\] \[(w + 2)^4 = w^4 + 8w^3 + 24w^2 + 32w + 16\] So, the expanded form using the binomial theorem is: \((w + 2)^4 = w^4 + 8w^3 + 24w^2 + 32w + 16\).

Key Concepts

Binomial Coefficient

When working with binomial expressions, the binomial coefficient is a crucial concept. It helps us determine the contribution of each term to the overall expression expansion. This coefficient is found in each component of a binomial expansion and originates from combinations found in probability and algebra.

The binomial coefficient \( \binom{n}{k} \) is defined as the number of ways to choose \( k \) elements from a set of \( n \) elements without regard to the order. Mathematically, it is represented as:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

Factorials (!) play a key role in calculating these coefficients, with \( n! \) representing the product of all positive integers less than or equal to \( n \).

For example, when expanding \((w + 2)^4\), we calculated:

• \( \binom{4}{0} = 1 \)
• \( \binom{4}{1} = 4 \)
• \( \binom{4}{2} = 6 \)
• \( \binom{4}{3} = 4 \)
• \( \binom{4}{4} = 1 \)

These coefficients are essential in determining the weight each term carries in the binomial expansion.

Algebraic Expansion

Algebraic expansion is the process of expressing a power of a binomial as a sum involving terms each of which is a product of a coefficient, a power of the first term, and a power of the second term. Using the binomial theorem, expansions involve breaking down expressions with exponents into simpler sum-of-products form.

Let's take the example of \((w + 2)^4\) and break it down:

• The formula we used is \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\).

In this scenario, \(a = w\), \(b = 2\), and \(n = 4\). This gives us five terms in total, each arising from applying the binomial coefficient to particular powers of \(w\) and \(2\). The complete expansion is composed by finding and simplifying each term individually:

• \(w^4\)
• \(8w^3\)
• \(24w^2\)
• \(32w\)
• \(16\)

This expansion process is vital in algebra, providing a more usable format of polynomial equations for solving and integration.

Factorial Calculation

Factorial computation is a recurring theme in mathematics, often cropping up in permutations, combinations, and the binomial theorem. A factorial, denoted by the symbol \(!\), is the product of all positive integers up to a given number.

It is defined mathematically as follows:

• \(n! = n \times (n-1) \times (n-2) \times \ldots \times 1\)

By convention, \(0!\) is defined as 1. Factorials are pivotal in calculating binomial coefficients as seen from their formula:

• \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).

In our example, computing \(4!\):

• \(4! = 4 \times 3 \times 2 \times 1 = 24\)

Factorials tend to grow very quickly, making manual computation somewhat cumbersome for larger numbers. Understanding factorials is essential for solving problems involving permutations, combinations, and binomial expansions seamlessly. This deep comprehension ensures calculating binomial coefficients becomes straightforward, aiding in clear and accurate algebraic expansion. This component, while small, is highly influential in solving and understanding such mathematical problems effectively.