Problem 65 - College Algebra

Question

Graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. $$ \begin{array}{l}f_{1}(x)=(x+1)^{4} \\\f_{2}(x)=x^{4} \\\f_{3}(x)=x^{4}+4 x^{3} \\\f_{4}(x)=x^{4}+4 x^{3}+6 x^{2} \\\f_{5}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x \\\f_{6}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x+1\end{array} $$ Use a \([-5,5,1]\) by \([-30,30,10]\) viewing rectangle.

Step-by-Step Solution

Answer

Each function graph corresponds to a different stage of the expansion of (x+1)^4, reflecting the stages of the Binomial Theorem. As new terms are added to the expansion, the graph of the function changes noting an extra term appearing, visually representing the progressive addition of terms in the Binomial Theorem expansion.

Step 1: Understand the Functions

We are given six functions \(f_{1}(x) = (x+1)^{4}\), \(f_{2}(x) = x^{4}\), \(f_{3}(x) = x^{4}+4 x^{3}\), \(f_{4}(x) = x^{4}+4 x^{3}+6 x^{2}\), \(f_{5}(x) = x^{4}+4 x^{3}+6 x^{2}+4 x\), and \(f_{6}(x) = x^{4}+4 x^{3}+6 x^{2}+4 x+1\).

Step 2: Calculating the Points for Plotting

The viewing rectangle is \([-5,5,1]\) by \([-30,30,10]\), so the x-values run from -5 to 5, and the y-values from -30 to 30. Replace the x in each function with values ranging from -5 to 5 and compute the corresponding y.

Step 3: Plotting the Functions

Plot each function into the same graph using the computed points. Each function should be assigned a unique color or symbol to distinguish them from each other on the plot.

Step 4: Relating to Binomial Theorem

Observe the graph. The way the functions build on one another in plotting mirrors how terms are added in each subsequent polynomial of the binomial expansion. Namely, \(f_{1}\) demonstrates base form of the binomial theorem; then each \(f_{2}, f_{3}, f_{4}, f_{5}\), and \(f_{6}\), respectively, adds another term of components in the expansion of \((x+1)^{4}\).

Key Concepts

Graphing Functions

Graphing functions is a powerful way to visualize mathematical relationships. It involves plotting points on a coordinate system, typically using an x-axis, representing the input values, and a y-axis, representing the output values. This method transforms abstract algebraic expressions into visible forms. When graphing the functions from the original problem, plot each function's computed points over the given range to differentiate their behavior.
For the functions given, use unique colors or plot styles to distinguish each one:

• f₁(x) : red
• f₂(x) : blue
• f₃(x) : green
• And so on...

Once the points are plotted, they form curves (or lines, in some cases) on the graph, offering insights into each function's nature and how they compare, just like in the given exercise.

Polynomial Functions

Polynomial functions are expressions that involve variables raised to integer powers, summed together, each with coefficients. They take the general form \[a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\] where \(a_n, a_{n-1}, ..., a_0\) are constants and \(n\) is a non-negative integer. In the problem's context, the functions are examples of polynomials, gradually building from \(x^4\) to \(x^4 + 4x^3 + 6x^2 + 4x + 1\).
These represent each stage of the binomial expansion.

• f₁(x) starts the sequence with just one term at the highest power.
• As you move to f₂(x), more terms join the function, increasing its complexity.

Recognizing these stages helps contextualize how binomial expansions unfold in polynomial terms.

Binomial Expansion

The binomial expansion is a method to expand expressions raised to a power into a sum of terms. It follows the general rule: \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] This is easily demonstrated using \((x + 1)^4\) represented by the functions in the exercise.

• f₁(x) : Represents the complete expression \((x + 1)^4\).
• Subsequent functions, like f₂(x), f₃(x), etc., build upon this as more terms are added.

The job of each function is to represent a partial sum, highlighting how each term contributes to the full expansion. This is visualized in stages in the original exercise, showing how polynomial functions link back to binomial expansions.

Binomial Coefficients

Binomial coefficients are crucial in expanding binomials. They appear in the binomial theorem as \( \binom{n}{k} \), commonly read as "n choose k," and represent the number of ways to choose \(k\) elements from \(n\) total elements. In the polynomial expansion, these coefficients multiply the terms in the expression, like \(x^4, 4x^3, 6x^2, 4x, \text{and} 1\) in our functions.

They form the framework of the entire expansion process:

• The first few coefficients \(1\), \(4\), \(6\), \(4\), and \(1\) are the components of the expansion for \((x + 1)^4\).
• They ensure each polynomial stage from f₂(x) to f₆(x) accurately sums to the complete expression.

Understanding these coefficients is paramount in mastering the binomial theorem, as seen in the illustrative graphs of the exercise.