Problem 41 - 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 \(f_{2}(x)\), such that \(f_{2}(x)\) is the sum of the fi st two terms of the expansion.

Step-by-Step Solution

Answer

\(f_2(x) = x^4 + 12x^3\); plotted as a cubic graph.

Step 1: Understand the Binomial Theorem

The Binomial Theorem provides a formula to expand expressions that are raised to a power. For a binomial of the form \((a + b)^n\), the expansion is given by \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). Here, \(\binom{n}{k}\) represents the binomial coefficient.

Step 2: Identify Terms for Expansion

In our problem, \(a = x\), \(b = 3\), and \(n = 4\). We will expand \((x + 3)^4\) using the binomial theorem.

Step 3: Calculate Binomial Coefficients

Calculate each binomial coefficient \(\binom{4}{k}\) for \(k = 0, 1, 2, 3, 4\). These are \(1, 4, 6, 4, 1\) respectively, calculated as follows: \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), \(\binom{4}{3} = 4\), \(\binom{4}{4} = 1\).

Step 4: Expand the Binomial

Substitute the values into the binomial expansion formula: \((x + 3)^4 = \binom{4}{0} x^{4} 3^{0} + \binom{4}{1} x^{3} 3^{1} + \binom{4}{2} x^{2} 3^{2} + \binom{4}{3} x^{1} 3^{3} + \binom{4}{4} x^{0} 3^{4}\).

Step 5: Simplify Each Term

Simplify each term in the expansion: - First term: \(1 \cdot x^4 \cdot 1 = x^4\)- Second term: \(4 \cdot x^3 \cdot 3 = 12x^3\)- Third term: \(6 \cdot x^2 \cdot 9 = 54x^2\)- Fourth term: \(4 \cdot x^1 \cdot 27 = 108x\)- Fifth term: \(1 \cdot 1 \cdot 81 = 81\)The full expansion is: \(x^4 + 12x^3 + 54x^2 + 108x + 81\).

Step 6: Find f2(x)

To find \(f_2(x)\), sum the first two terms of the expansion: \(f_2(x) = x^4 + 12x^3\).

Step 7: Graph f2(x)

Graph the polynomial \(f_2(x) = x^4 + 12x^3\). Use a set of axes and plot several points by evaluating \(f_2(x)\) for different values of \(x\) to see the shape of the graph. This is a cubic polynomial with no constant or linear terms, so it will have a turning point.

Key Concepts

Binomial Expansion

Binomial expansion is a method used to expand expressions that contain two terms, raised to a power, into a sum of terms. This concept is derived from the Binomial Theorem, which greatly simplifies the process of handling expressions like \((a + b)^n\).
This theorem tells us that any binomial raised to a power can be expressed as a sum: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]

• \(n\) is the power to which the binomial is raised.
• \(a\) and \(b\) are the terms in the binomial.
• \(k\) is a term counter for each part of the sum.

The theorem unfolds the binomial into multiple terms containing coefficients (which are, themselves, numbers) that help in forming the complete expansion. By breaking it down this way, complicated polynomial operations become more manageable.

Binomial Coefficients

Binomial coefficients are key parts of the binomial expansion formula. They determine the number of ways each term in the expansion is formed. A binomial coefficient is written as \(\binom{n}{k}\), where \(n\) represents the total number of terms in the binomial, and \(k\) represents the specific term in the sequence.
These coefficients can be found using combinations, often through this formula: \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\]

• \(!\) stands for factorial, which is the product of all positive integers up to that number.

For example, in expanding \((x + 3)^4\), the coefficients are determined as \(1, 4, 6, 4, 1\). These numbers indicate how each expanded term contributes to creating the overall polynomial.

Cubic Polynomial

A cubic polynomial is a polynomial of degree three, typically associated with expressions like \(ax^3 + bx^2 + cx + d\).< br>In this context, "cubic" relates to the highest degree present in the polynomial, which is three, meaning the variable is raised to the third power in one term. Cubic polynomials can have different shapes, depending on their coefficients.
For example, in the problem, the focus is on finding \(f_2(x)\), the sum of the first two terms of the binomial expansion of \((x + 3)^4\), which yields \(x^4 + 12x^3\).
Here, even though the highest degree seems to be four, the presence of significant third-degree effects makes it exhibit behaviors often linked to cubic polynomials, particularly due to the large coefficient, \(12\), in front of \(x^3\). Graphing these can reveal turning points, and curves inherent in such functions.