Problem 67 - College Algebra

Question

Use the Binomial Theorem to find a polynomial expansion for each function. Then use a graphing utility and an approach similar to the one in Exercises 64 and 65 to verify the expansion. $$ f_{1}(x)=(x-2)^{4} $$

Step-by-Step Solution

Answer

The binomial expansion for \(f_{1}(x) = (x-2)^{4}\) is \( x^4 - 8x^3 + 24x^2 - 32x + 16 \). The graphing utility would show both graphs as identical, verifying the expansion.

Step 1: Using the Binomial Theorem

Using the Binomial Theorem, which states that \((x - y)^n = \sum_{k=0}^{n}(x^{n - k}y^k)\), the binomial can be expanded. Thus, \( (x - 2)^4 = x^4 - 4x^3 * 2 + 6x^2 * 2^2 - 4x * 2^3 + 2^4\)

Step 2: Simplifying the expression

Simplify the expression to obtain the polynomial. So, \(x^4 - 4x^3 * 2 + 6x^2 * 2^2 - 4x * 2^3 + 2^4\) becomes \(x^4 - 8x^3 + 24x^2 - 32x + 16\)

Step 3: Verifying the expansion by graphing

To verify the results, plot both the original function \(f_{1}(x) = (x-2)^{4}\) and the expanded polynomial \(x^4 - 8x^3 + 24x^2 - 32x + 16\). If they are identical, then the Binomial Theorem has been applied correctly. Remember to label both graphs clearly.

Key Concepts

Polynomial Expansion

Understanding polynomial expansion is vital when working with powers of binomials like in the exercise with the function \( f_{1}(x)=(x-2)^{4} \). Polynomial expansion is the process of converting a power of a binomial into a sum of terms called a polynomial, which involves terms with descending powers of the variable. The Binomial Theorem is a powerful shortcut to finding these polynomials without having to multiply the binomial by itself several times.

Using this theorem, we can expand \( (x - y)^n \) for a particular power \( n \) in terms of \( x \) and \( y \) without direct multiplication. The expansion of \( (x - y)^n \) results in a polynomial where the coefficient of each term is a binomial coefficient. These coefficients correspond to the number of ways to choose a certain number of items from a larger set and are often represented as \( \binom{n}{k} \), which reads as 'n choose k'. For example, for \( (x - 2)^4 \) as shown in the exercise, the expanded form results in \( x^4 - 8x^3 + 24x^2 - 32x + 16 \) after simplification.

While the step-by-step solution provided goes through the expansion, it's crucial to understand that each term of the expanded polynomial involves these binomial coefficients and powers of \( x \) and \( -2 \) that correspond to the original binomial.

Graphing Utility

A graphing utility is an essential tool for visualizing mathematical functions, confirming algebraic calculations, and improving comprehension of mathematical concepts. In the context of polynomial expansion, a graphing utility can be used to confirm the equivalence of the original function and the expanded polynomial.

By entering both the original function \( f_{1}(x) = (x-2)^{4} \) and the expanded form \( x^4 - 8x^3 + 24x^2 - 32x + 16 \) into a graphing utility, we can compare them visually. If their graphs overlap perfectly, it serves as strong evidence that our polynomial expansion is correct.

When using a graphing utility, it's important to properly set the range and scale of the graph to ensure that all relevant points, such as intercepts and turning points, are visible. Labeling is also critical to distinguish between the graphs of the original function and the expansion. This visual confirmation reinforces understanding and provides a practical application of polynomial expansion theory.

Algebraic Expression Simplification

Algebraic expression simplification is the process of reducing a complex algebraic expression into its simplest form. This involves combining like terms, reducing fractions, and applying properties of exponents and coefficients. For example, as part of the process in the exercise, we simplify the expansion of \( (x - 2)^4 \) by combining like terms and multiplying coefficients and powers to achieve the simplified polynomial \( x^4 - 8x^3 + 24x^2 - 32x + 16 \).

Simplifying expressions makes them easier to work with, both algebraically and graphically. When an expression is simplified, it is more readily analyzed, and attributes such as intercepts, end behavior, and concavity become clearer. Additionally, simplified expressions are less prone to errors when computing derivatives or integrals if needed in further steps.

As the final part of the exercise, after simplifying the polynomial expansion, it's advisable to recheck each term to verify that all coefficients are accurate, which ensures the simplified polynomial precisely represents the expanded form of the original function. This attention to detail guarantees the most fundamental part of the exercise is error-free, leading to a successful verification through graphing.