Problem 40 - College Algebra

Question

For the following exercises, use the Binomial Theorem to expand the binomial \(f(x)=(x+3)^{4} .\) Then find and graph each indicated sum on one set of axes. Find and graph \(f_{1}(x)\), such that \(f_{1}(x)\) is the fi st term of the expansion.

Step-by-Step Solution

Answer

The first term in the expansion is \(x^4\), and it's graph is a steep parabola opening upwards.

Step 1: Understand the Binomial Theorem

The Binomial Theorem provides a way to expand expressions in the form \((a + b)^n\). It states: \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(a = x\), \(b = 3\), and \(n = 4\).

Step 2: Identify Binomial Coefficient for First Term

To find the first term in the expansion, look at the case where \(k = 0\). According to the Binomial Theorem, the coefficient of \(a^{n-k}\) is given by \(\binom{n}{k}\).

Step 3: Calculate the First Term of the Expansion

Using \(k = 0\), calculate the first term as follows:\[y = \binom{4}{0} x^{4-0} 3^0 = 1 \cdot x^4 \cdot 1 = x^4\]. Thus, the first term is \(x^4\).

Step 4: Graph the First Term

Since \(f_1(x) = x^4\), plot this function on a set of axes. It's a simple power function with a graph that resembles a parabola, opening upwards, but much steeper than \(x^2\).

Key Concepts

Binomial Expansion

The binomial expansion is a powerful algebraic tool that helps in expanding expressions raised to a power. This is particularly useful for binomials like \((x + 3)^4\). The Binomial Theorem states that:

• \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

Here, you can see it uses the concept of summation (indicated by the sigma \(\sum\) symbol) and runs from \(k=0\) to \(n\).
In this theorem:

• \(a = x\)
• \(b = 3\)
• \(n = 4\)

The Binomial Theorem provides the complete expansion, offering a strong foundation for understanding polynomial expansions. It works by breaking down complex expressions into simpler terms, which can be easily calculated.

Binomial Coefficient

A crucial part of the binomial expansion is the binomial coefficient, denoted as \(\binom{n}{k}\). This is used to determine the weight or importance of each term in the expansion. The binomial coefficient \(\binom{n}{k}\) is calculated using the formula:

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

Where \(n!\) is the factorial of \(n\).
For example, in expanding \((x + 3)^4\), we use the binomial coefficient to find the value of the very first term. By setting \(k=0\), we use:

• \(\binom{4}{0} = \frac{4!}{0!4!} = 1\)

Hence, the first term in the expansion is \(1 \cdot x^4 \cdot 1 = x^4\). Binomial coefficients help construct each term of the expansion systematically and accurately.

Graphing Polynomial Functions

Graphing polynomial functions, such as \(f_1(x) = x^4\), is an essential skill in understanding behavior and properties of equations. The function \(x^4\) is a higher-degree monomial compared to \(x^2\) and is similar in appearance to a parabola but steeper.
Key characteristics of the graph of \(x^4\) are:

• It resembles a U-shaped graph, opening upwards.
• It is symmetric about the y-axis, indicating it is an even function.
• As \(x\) approaches zero, the graph remains close to the x-axis. Conversely, as \(x\) moves away from zero, the y-values increase sharply.

To graph such functions:

• Choose a range of \(x\) values, including both negative and positive.
• Calculate the corresponding \(y\) values using the function formula.
• Plot each point on the graph and draw a smooth curve through them.

Understanding polynomial graphs helps us visually interpret the behavior and turning points of functions. Plotting them makes mathematical concepts tangible and easier to grasp.