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

Step-by-Step Solution

Answer

The expanded polynomial is \((x+3)^4 = x^4 + 12x^3 + 54x^2 + 108x + 81\). The first three terms are \(x^4 + 12x^3 + 54x^2\).

Step 1: Recall the Binomial Theorem

The Binomial Theorem allows us to expand a binomial expression of the form \((a + b)^n\). The theorem states: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] where \(\binom{n}{k}\) is the binomial coefficient calculated as \(\frac{n!}{k!(n-k)!}\).

Step 2: Set up for Expansion

Identify \(a = x\), \(b = 3\), and \(n = 4\) in the expression \((x+3)^4\). We want to expand this using the Binomial Theorem.

Step 3: Calculate Binomial Coefficients

Calculate each coefficient \(\binom{4}{k}\) for \(k = 0, 1, 2, 3, 4\): - \(\binom{4}{0} = 1\)- \(\binom{4}{1} = 4\)- \(\binom{4}{2} = 6\)- \(\binom{4}{3} = 4\)- \(\binom{4}{4} = 1\)

Step 4: Expand Using the Binomial Theorem

Substituting into the binomial expansion formula, we have: \[(x+3)^4 = \binom{4}{0}x^4 + \binom{4}{1}x^3 \cdot 3 + \binom{4}{2}x^2 \cdot 3^2 + \binom{4}{3}x^1 \cdot 3^3 + \binom{4}{4} \cdot 3^4\] This simplifies to: \[x^4 + 12x^3 + 54x^2 + 108x + 81\]

Step 5: Identify the First Three Terms

The first three terms of the expansion are:\[f_3(x) = x^4 + 12x^3 + 54x^2\]

Step 6: Graph the First Three Terms

To graph \(f_3(x) = x^4 + 12x^3 + 54x^2\), plot the polynomial for different values of \(x\), typically from -10 to 10, to see the behavior of the function. Note the shape of the polynomial and key features like intercepts and curvature.

Key Concepts

Binomial Expansion

The binomial expansion allows us to expand expressions that are raised to a power. In mathematics, the Binomial Theorem provides a formula for expanding expressions of the form \((a + b)^n\). This means that you can take a squared, cubed or any higher power of a binomial and expand it into a polynomial. The formula for this expansion is given by:

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

where \(\binom{n}{k}\) are known as the binomial coefficients. This formula breaks down the process into individual terms, each involving powers of \(a\) and \(b\), multiplied by the respective binomial coefficients. When solving the given exercise, you identify each part of the binomial expression \((a = x, b = 3, n = 4)\) and substitute it into the formula, enabling us to expand \((x+3)^4\) smoothly. The expansion yields a polynomial: \(x^4 + 12x^3 + 54x^2 + 108x + 81\). Remember, binomial expansion is especially handy for calculating high powers of binomials, which would otherwise be tedious to compute through distributive methods.

Polynomial Graphing

Polynomial graphing is essential when visualizing the behavior of polynomial equations across different values of \(x\). To practically apply polynomial graphing, once you have expanded a binomial using the Binomial Theorem, you can plot the resulting polynomial. In the exercise, we focus on \(f_3(x) = x^4 + 12x^3 + 54x^2\), the first three terms of our expansion.Graphing provides insight into the function's characteristics:

• It shows where the polynomial crosses the x-axis (roots or zeros) and the y-axis (y-intercepts).
• It depicts the overall shape or curvature of the function, which is useful for understanding the polynomial's slope and its increasing or decreasing behavior.
• The graph also highlights turning points or local maxima and minima.

To graph \(f_3(x)\), choose values for \(x\), such as from \(-10\) to \(10\), and plot the resulting \(y\)-values on a coordinate plane. Utilizing software or a graphing calculator can make this process more efficient and accurate compared to manual plotting.

Binomial Coefficients

The binomial coefficients \(\binom{n}{k}\) play a crucial role in the binomial expansion and are essential for determining the specific terms of a polynomial resulting from such an expansion. These coefficients are derived from combinations and are calculated using the formula:

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

Here, \(n!\) denotes factorial notation, meaning "n factorial," which is the product of all positive integers up to \(n\). In our exercise with \((x+3)^4\), calculating the binomial coefficients involves substituting for \(n = 4\) and \(k = 0, 1, 2, 3, 4\). Computations track as:

• \(\binom{4}{0} = 1\)
• \(\binom{4}{1} = 4\)
• \(\binom{4}{2} = 6\)
• \(\binom{4}{3} = 4\)
• \(\binom{4}{4} = 1\)

The value of each coefficient determines how much each term contributes to the overall polynomial. These coefficients reflect the number of ways to choose \(k\) items from \(n\) options, which in turn influences the symmetry and distribution of terms within the expanded binomial expression.