Problem 69 - College Algebra

Question

In Exercises 68-69, graph each of the functions in the same viewing rectangle. Describe how the graphs illustrate the Binomial Theorem. \(f_{1}(x)-(x+1)^{4}\) \(f(x)=x^{4}\) \(f_{3}(x)-x^{4}+4 x^{3} \quad f_{4}(x)-x^{4}+4 x^{3}+6 x^{2}\) \(f_{s}(x)-x^{4}+4 x^{3}+6 x^{2}+4 x\) \(f_{\phi}(x)-x^{4}+4 x^{3}+6 x^{2}+4 x+1\) Use a \([-5,5,1]\) by \([-30,30,10]\) viewing rectangle.

Step-by-Step Solution

Answer

The graphs of functions provide a visual representation of the Binomial theorem. As each term is added to the expansion, the graph of the function changes, illustrating the way each term affects the sum. The last graph, which includes all terms, gives a final and complete shape of the binomial expansion, showing the combined effect of all terms.

Step 1: Recognize the Expanded Form

First, observe and recognize that \(f_{1}(x), f_{2}(x), f_{3}(x), f_{4}(x), f_{5}(x)\) are the incremental expansions of \((x+1)^{4}\) in terms of x: \(f_{1}(x)-(x+1)^{4}\)\(f_{2}(x)=x^{4} = (x+1)^{4} - 4*(x+1)^{3}\)\(f_{3}(x)=x^{4}+4 x^{3} = (x+1)^{4} - 4*(x+1)^{3}+6*(x+1)^{2}\)\(f_{4}(x)=x^{4}+4 x^{3}+6 x^{2} = (x+1)^{4} - 4*(x+1)^{3}+6*(x+1)^{2} - 4*(x+1)\)\(f_{5}(x)=x^{4}+4 x^{3}+6 x^{2}+4 x+1 = (x+1)^4\)

Step 2: Plot the Graphs

Now, plot the graphs of these functions. Use a \([-5,5,1]\) by \([-30,30,10]\) viewing rectangle to accommodate the range of these functions.

Step 3: Analyze the Graphs

Once drawn, observe how the graphs illustrate the Binomial theorem. As each term is added, the graph gains complexity, illustrating the gradual construction of the binomial expansion. The endpoint, \(f_{5}\), corresponds to the full expansion of \((x+1)^{4}\), thus visually demonstrating the Binomial theorem.

Key Concepts

Graphing Polynomial Functions

Graphing polynomial functions involves plotting points that represent solutions to the polynomial equations. This helps us visualize the behavior of the function across different values of \(x\).
By using specified viewing windows such as \([-5,5,1]\) by \([-30,30,10]\), we ensure that the most relevant points are visible.

• Visual patterns, like curves and lines, become evident, displaying features like intercepts and turning points.
• In this exercise, each graph illustrates a step in the binomial expansion of \((x+1)^4\), showing the complexity of the polynomial as terms are added.

Binomial Expansion

The Binomial Theorem provides a way to expand expressions that involve powers of a binomial, such as \((x+1)^n\).
In this exercise, we focused on \((x+1)^4\), and visualized its incremental expansions through various functions.

• Each function represents an additional term being added to the expansion.
• The complete expansion results in a polynomial where all the terms of the binomial are expressed in algebraic form.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators. They form the building blocks for polynomials and are crucial for the binomial expansion.
For this problem, you encounter expressions like \(x^4\), \(4x^3\), etc.

• Understanding how to manipulate these expressions is crucial for expanding binomials and solving polynomials effectively.
• Each expansion step in the exercise involved adding these expressions incrementally to build the full polynomial.

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers in one or more variables multiplied by coefficients. The degree of the polynomial is determined by the highest power of \(x\).
In our example, \(x^4\) is the highest term.

• These functions can be used to model various real-world scenarios, such as physics and engineering problems.
• Graphing these functions shows how they behave over a range of values and helps compare different polynomial equations.

Decomposition of Polynomials

Decomposing polynomials involves breaking down a complex polynomial into simpler components. This approach is used to simplify expressions and solve polynomial equations efficiently.
The exercise demonstrates decomposition through incremental addition of terms.

• Starting from simpler polynomials, each additional term increases complexity, allowing a deeper understanding of the function's behavior through its expansion.
• This can be useful in factoring and simplifying polynomials to solve equations more easily.