Problem 19 - Algebra for College Students

Question

Use the binomial theorem to expand each binomial. $$(x+2 a)^{3}$$

Step-by-Step Solution

Answer

(x+2a)^3 = x^3 + 6ax^2 + 12a^2x + 8a^3.

Step 1 - Understand the Binomial Theorem

The Binomial Theorem states that \[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \] where \( \binom{n}{k} \) are the binomial coefficients. In this case, \( x = x \), \( y = 2a \), and \( n = 3 \).

Step 2 - Identify Binomial Coefficients

Calculate the binomial coefficients using \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). For \( n = 3 \), the coefficients are: \[ \binom{3}{0} = 1, \binom{3}{1} = 3, \binom{3}{2} = 3, \binom{3}{3} = 1. \]

Step 3 - Apply the Binomial Theorem

Substitute \( n = 3 \), \( x \), and \( 2a \) into the Binomial Theorem formula: \[ (x+2a)^3 = \binom{3}{0} x^3 (2a)^0 + \binom{3}{1} x^2 (2a)^1 + \binom{3}{2} x^1 (2a)^2 + \binom{3}{3} x^0 (2a)^3. \]

Step 4 - Simplify Each Term

Evaluate and simplify each term: \[ = 1 \cdot x^3 \cdot 1 + 3 \cdot x^2 \cdot 2a + 3 \cdot x \cdot (2a)^2 + 1 \cdot 1 \cdot (2a)^3 \]

Step 5 - Final Expression

Combine all simplified terms: \[ (x+2a)^3 = x^3 + 6ax^2 + 12a^2x + 8a^3 \]

Key Concepts

Binomial Coefficients

To understand the expansion of binomials, we first need to grasp the concept of binomial coefficients. Binomial coefficients, often denoted as \( \binom{n}{k} \), represent the number of ways to choose \( k \) elements from a set of \( n \) elements. It's often seen in combinatorics and is called a 'combination'.
The formula for binomial coefficients is: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]
Here, \( n! \) (factorial) means the product of all positive integers up to \( n \). We will cover factorials in more detail later. To make it clear, if you need to compute \( \binom{3}{1} \) for our exercise, you plug in the values: \( n = 3 \) and \( k = 1 \), resulting in \( \frac{3!}{1!(3-1)!} = \frac{3 \times 2 \times 1}{1 \times 2 \times 1} = 3 \).
Binomial coefficients are very important as they help determine the terms in the expansion when we use the Binomial Theorem.

Polynomial Expansion

The Binomial Theorem is a powerful tool for polynomial expansion. The theorem states: \[ (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \]
Here, \( x \) and \( y \) are any terms, and \( n \) is a non-negative integer. The expansion results in a series of terms where each term is a combination of powers of \( x \) and \( y \), scaled by the binomial coefficients.
For our example \( (x + 2a)^3 \), using \( x = x \), \( y = 2a \), and \( n = 3 \), the polynomial expansion follows these steps:
1. Identify terms with coefficients: \[ \binom{3}{0} x^3 (2a)^0 + \binom{3}{1} x^2 (2a)^1 + \binom{3}{2} x (2a)^2 + \binom{3}{3} (2a)^3 \]
2. Substitute the known coefficients: \[ = 1 \cdot x^3 \cdot 1 + 3 \cdot x^2 \cdot 2a + 3 \cdot x \cdot (2a)^2 + 1 \cdot 1 \cdot (2a)^3 \]
3. Simplify each term: \[ = x^3 + 6ax^2 + 12a^2x + 8a^3 \]
This is the expanded form of our binomial.

Factorials

Factorials are a fundamental concept in mathematics used in permutations, combinations, and our current context of binomial coefficients. The factorial of a number \( n \), denoted \( n! \), is the product of all positive integers up to and including \( n \).
For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Similarly, \( 0! = 1 \) by definition, a useful property in combinatorial calculations.
Factorials describe the total number of ways to arrange \( n \) objects in order. They're pivotal in calculating binomial coefficients.
For instance, in our exercise, to compute \( \binom{3}{1} \):
1. Calculate the numerator: \( 3! = 3 \times 2 \times 1 = 6 \)
2. Calculate the denominators separately: \( 1! = 1 \) and \( (3-1)! = 2 \times 1 = 2 \)
3. Combine using the formula: \[ \binom{3}{1} = \frac{3!}{1!(3-1)!} = \frac{6}{2} = 3 \]
Understanding factorials helps simplify and clarify the steps involved in using the Binomial Theorem.