Problem 42 - 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 \(f_{3}(x)\), such that \(f_{3}(x)\) is the sum of the fi st three terms of the expansion.

Step-by-Step Solution

Answer

The first three terms are \( x^4 + 12x^3 + 54x^2 \), represented by \( f_3(x) \).

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that for any integer \( n \) and any numbers \( a \) and \( b \), the expansion of \( (a+b)^n \) can be written as \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). For this problem, \( a = x \), \( b = 3 \), and \( n = 4 \).

Step 2: Compute the Binomial Coefficients

The binomial coefficients are computed using the formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \). For \( n = 4 \), the coefficients \( \binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3} \) are 1, 4, 6, and 4 respectively.

Step 3: Expand the Binomial

Using the coefficients calculated, and substituting \( a = x \) and \( b = 3 \), the expansion is: \( (x+3)^4 = \binom{4}{0} x^4 + \binom{4}{1} x^3(3) + \binom{4}{2} x^2(3^2) + \binom{4}{3} x(3^3) + \binom{4}{4}(3^4) \).

Step 4: Simplify the Expression

Substituting the binomial coefficients and simplifying term by term gives: \( x^4 + 12x^3 + 54x^2 + 108x + 81 \).

Step 5: Determine \( f_3(x) \)

The function \( f_3(x) \) represents the sum of the first three terms of the expansion. Thus, \( f_3(x) = x^4 + 12x^3 + 54x^2 \).

Step 6: Graph \( f_3(x) \) and the Original Expansion

Graph both \( f(x) = (x+3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81 \) and \( f_3(x) = x^4 + 12x^3 + 54x^2 \) on the same set of axes to compare the sums visually.

Key Concepts

Binomial Coefficients

The Binomial Theorem is a powerful tool in algebra that allows us to expand expressions of the form \((a+b)^n\). Central to this theorem are the binomial coefficients, which dictate the number of ways to select elements from a set. These coefficients can be computed using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), where \(n!\) (n factorial) is the product of all positive integers up to \(n\).
For the expansion of \((x+3)^4\), we specifically need the coefficients: \(\binom{4}{0}\), \(\binom{4}{1}\), \(\binom{4}{2}\), and \(\binom{4}{3}\), which evaluate to 1, 4, 6, and 4 respectively. These coefficients tell us how many times each term in the polynomial will appear, forming the weighted sum of the expansion.
Understanding these coefficients is crucial, as they allow you to predict how powerful each component of the binomial expansion will be, essentially shaping the polynomial's term structure.

Polynomial Expansion

Polynomial expansion using the Binomial Theorem involves transforming a binomial expression into a sum of terms raised to different powers. Consider \((x+3)^4\). By applying the binomial expansion formula, each term is computed as \(\binom{n}{k} a^{n-k} b^k\). This means you calculate each term using the binomial coefficient and the powers of the variables involved.
For instance, in \((x+3)^4\):

• The first term \(\binom{4}{0} x^4 = 1 \cdot x^4\)
• The second term \(\binom{4}{1} x^3 \cdot 3 = 4 \cdot 3x^3 = 12x^3\)
• The third term \(\binom{4}{2} x^2 \cdot 3^2 = 6 \cdot 9x^2 = 54x^2\)
• And so on for each subsequent term.

By expanding, we convert the binomial into a polynomial, capturing its behavior as a sum of multiple terms, each with different degrees and coefficients.

Algebraic Functions

Algebraic functions, like \(f(x) = (x+3)^4\), can be analyzed effectively using tools from algebra such as the Binomial Theorem. The expression \((x+3)^4\) doesn't initially reveal its structure until expanded into a polynomial form. This expansion – calculated step by step using the binomial coefficients – provides insights into the algebraic nature of the function.
Analyzing parts of the polynomial, such as the sum of its first three terms \(f_3(x) = x^4 + 12x^3 + 54x^2\), allows us to comprehend how collective terms behave. This subset of the function helps in understanding the weight of each term towards the overall shape of the graph.

• Such algebraic functions are important in graphing, as each term influences the way the function curves and the nature of its slopes.
• Graphing these helps in visualizing the differences and the overall trajectory of polynomial behavior over its domain.

Algebraic functions thus not only tell us about individual terms but also illustrate the collective influence on the polynomial's graph and overall behavior.