Problem 41 - College Algebra with Corequisite Support

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 first two terms of the expansion.

Step-by-Step Solution

Answer

Expand \((x+3)^4\) to find \(f_2(x) = x^4 + 12x^3\) and graph this polynomial.

Step 1: Understanding the Binomial Theorem

The Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\), where \(\binom{n}{k}\) is the binomial coefficient. In this exercise, we need to expand \((x+3)^4\).

Step 2: Determine the Binomial Coefficients

Calculate each binomial coefficient using \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) for \(n=4\). They are: \(\binom{4}{0} = 1\), \(\binom{4}{1} = 4\), \(\binom{4}{2} = 6\), \(\binom{4}{3} = 4\), \(\binom{4}{4} = 1\).

Step 3: Write the Full Expansion Using the Binomial Theorem

Substitute the values \(a = x\), \(b = 3\), and \(n = 4\) into the binomial theorem: \((x+3)^4 = \sum_{k=0}^{4} \binom{4}{k} x^{4-k} 3^k\).

Step 4: Calculate Each Term of the Expansion

Calculate each term: - For \(k=0\), the term is \(\binom{4}{0} x^4 \cdot 3^0 = x^4\). - For \(k=1\), the term is \(\binom{4}{1} x^3 \cdot 3^1 = 12x^3\). - Remaining terms can be calculated but are not needed for \(f_2(x)\).

Step 5: Identify \(f_2(x)\) from the Expansion

Since \(f_2(x)\) is the sum of the first two terms: \(f_2(x) = x^4 + 12x^3\).

Step 6: Graph \(f_2(x)\) on the Axes

Plot the function \(f_2(x) = x^4 + 12x^3\) on a set of axes, showing the curve as a polynomial of degree 4.

Key Concepts

Binomial Expansion

The Binomial Expansion is a powerful algebraic tool used to expand expressions that are raised to a power. When we have an expression like \((x+3)^4\), expanding it means rewriting it as a sum of terms rather than keeping it compact as a binomial raised to a power. This allows us to see each term individually and understand how they contribute to the overall expression.
To expand \((a+b)^n\) using the Binomial Theorem, we use the formula: \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\).
In simpler terms:

• "\(a\)" and "\(b\)" are the terms of the binomial you want to expand.
• "\(n\)" is the power to which the binomial is raised.
• "\(\binom{n}{k}\)" is the binomial coefficient for each term in the expansion.
• The exponent of "\(a\)" decreases from \(n\) to 0, while the exponent of "\(b\)" increases from 0 to \(n\).

Understanding how to perform a binomial expansion can help in visualizing higher-degree polynomial functions and in simplifying computational problems.

Binomial Coefficients

When expanding a binomial using the Binomial Theorem, binomial coefficients play an essential role. They are the numbers found in Pascal's Triangle and determine the weight of each term in the expansion. In the formula \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\), these coefficients are denoted as "\(\binom{n}{k}\)".
To calculate the binomial coefficients:

• Use the formula: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
• "\(n!\)" (n factorial) means multiplying all integers from 1 to \(n\).
• "\(k\)" is the term number you are calculating the coefficient for.

For example, when expanding \((x+3)^4\), \(\binom{4}{0}\) is the coefficient for the first term, which is \(1\). \(\binom{4}{1}\) is for the second term, equal to \(4\), and so on.
These coefficients ensure that each term of the binomial expansion is correctly proportioned to respect the original binomial. They affect the size of each corresponding term, which is crucial when constructing polynomial functions.

Polynomial Functions

Polynomial functions consist of sums of terms like \(ax^n\), where \(a\) is a constant and \(n\) is a non-negative integer. Each term in the expanded form of a binomial expression like \((x+3)^4\) is a polynomial.
The expansion of \((x+3)^4\) results in a polynomial composed of several terms:

• \(x^4\), \(12x^3\)
• The remaining terms are obtained from further calculations but can be summarized within the expansion.

In mathematics, understanding polynomial functions is essential:

• They are used extensively in modeling real-world situations and solving diversified mathematical problems.
• Seeing them graphically helps to understand their behavior, such as finding roots, intercepts, and turning points.

For the exercise, identifying \(f_2(x) = x^4 + 12x^3\) efficiently initializes working with polynomials. Graphing it shows the real-world application and visual behavior of polynomial functions, illustrating how each term's contribution affects the curve.