Problem 16 - College Algebra

Question

Use the Binomial Theorem to expand each binomial and express the result in simplified form. $$ (3 x+1)^{4} $$

Step-by-Step Solution

Answer

The expansion of the binomial \((3x + 1)^4\) using Binomial Theorem is: \( 81x^4 + 108x^3 + 54x^2 + 12x +1 \)

Step 1: Identify the Variables

In the binomial \((3x + 1)^4\), \(a = 3x\), \(b = 1\), and \(n = 4\).

Step 2: Apply the Binomial Theorem

Applying the Binomial Theorem \((a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k}b^k \), we get \( (3x + 1)^4 = {4 \choose 0}(3x)^4*1^0+{4 \choose 1}(3x)^3*1^1+{4 \choose 2}(3x)^2*1^2+{4 \choose 3}(3x)^1*1^3+{4 \choose 4}(3x)^0*1^4 \)

Step 3: Simplify the Expression

Simplify the powers and binomial coefficients. After simplification, the equation should look like this: 81x^4 + 108x^3 + 54x^2 + 12x +1.

Key Concepts

Understanding Binomial Expansion

Binomial expansion is a technique used to expand expressions that are raised to a power and take the form of \((a + b)^n\). This method involves breaking down the expression into a series of terms. Each term consists of products of the form \(a^{n-k}b^k\), where \(n\) denotes the exponent, and the values \(k\) vary from 0 to \(n\).
In the expression \((3x + 1)^4\), this expansion means we'll have terms like \((3x)^{4-k} \, 1^k\) for different values of \(k\).
Here's a friendly recap of the process:

• The expression consists of two parts, \(3x\) and \(1\), called the binomial terms.
• We expand them by increasing the power \(k\) of \(1\) and decreasing the power of \(3x\) from the highest down to zero.

This breakdown facilitates handling polynomial expressions conveniently by distributing the powers across each term.

Role of Combinatorial Coefficients

Combinatorial coefficients, also known as binomial coefficients, play a crucial role in binomial expansions. They are represented by \({n \choose k}\) and computed as \(\frac{n!}{k!(n-k)!}\), where \(n!\) is the factorial of \(n\). These coefficients determine the weight or magnitude of each term in the expansion.
In the context of the problem \((3x + 1)^4\), each term in the expanded form has its own coefficient.
For example:

• For \(k = 0\), the coefficient is \({4 \choose 0} = 1\).
• For \(k = 1\), the coefficient is \({4 \choose 1} = 4\).
• For \(k = 2\), the coefficient is \({4 \choose 2} = 6\).
• Continuing in this fashion for the other terms in the expansion.

Understanding these coefficients is essential for achieving correct results when performing binomial expansions.

Simplifying Polynomials

Polynomial simplification involves combining like terms and reducing expressions to their simplest form. This process is crucial for making complex polynomial expressions manageable and understandable.
In the expansion of \((3x + 1)^4\), the expression yielded multiple terms that include powers of \(3x\). These terms were initially expanded and written as separate terms such as \(81x^4\), \(108x^3\), etc.
Simplification steps typically involve:

• Calculating and inserting values for powers of terms, e.g., \((3x)^3 = 27x^3\).
• Rearranging and combining like terms, which are terms having the same variable raised to the same power.
• In this example: the expression \((3x+1)^4\) simplifies to \(81x^4 + 108x^3 + 54x^2 + 12x + 1\), ensuring the polynomial is presented neatly.

This simplification makes the polynomials easier to work with for various algebraic applications.