Problem 66 - 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. $$ f_{1}(x)=(x-1)^{3} $$

Step-by-Step Solution

Answer

The polynomial expansion of function \(f_{1}(x) = (x-1)^{3}\) is \(f_{1}(x)=x^{3} - 3x^{2} + 3x - 1\). This can be verified graphically.

Step 1: Apply the Binomial Theorem

The Binomial Theorem states that for any numbers a and b, the equation \((a+b)^n = \sum _{k=0} ^{n} \binom{n}{k} a^{n-k} b^{k} \) holds true. Applying this to the given function \(f_{1}(x) = (x-1)^{3}\), we let a=x, b=-1 and n=3, which gives us the expansion.

Step 2: Calculate the Expansion

Applying the formula, we get \[f_{1}(x)=\binom{3}{0} x^{3} (-1)^{0} + \binom{3}{1} x^{2} (-1)^{1} + \binom{3}{2} x^{1} (-1)^{2} + \binom{3}{3} x^{0} (-1)^{3}\] This simplifies to \[f_{1}(x)=x^{3} - 3x^{2} + 3x - 1\]. This is the polynomial expansion of the function.

Step 3: Graphically Verify the Expansion

To graphically verify the expansion, plot both \(f_{1}(x)\) and the expanded polynomial on the same set of axes. If the expansion is correct, the two functions will coincide exactly.

Key Concepts

Polynomial Expansion

When we talk about polynomial expansion, we're referring to the process of expressing a power of a binomial, that is, \( (a + b)^n \), as a sum of terms involving different powers of \( a \) and \( b \), multiplied by specific coefficients. The Binomial Theorem is a fantastic tool that provides a systematic method for this type of expansion.

For example, if we want to expand \( (x-1)^3 \), we apply this theorem to break it down into a polynomial with several terms. The step-by-step method described in the textbook solution involves identifying the values of \( a \) and \( b \) from the binomial and substituting them into the binomial formula. In this case, \( a = x \) and \( b = -1 \), giving us the expanded form after applying the coefficients and powers according to the formula. The result is a clearer picture of how the original expression will behave for various values of \( x \) without performing tedious multiplication.

Binomial Coefficients

The numbers that appear as multipliers in the terms of the expanded polynomial, known as binomial coefficients, play a pivotal role in polynomial expansions. These coefficients are usually written as \( \binom{n}{k} \), where \( n \) is the power of the binomial, and \( k \) is the specific term number of the expansion.

Understanding the Coefficients

The binomial coefficient \( \binom{n}{k} \) represents the number of ways to choose \( k \) elements out of a set of \( n \) elements, without regard to the order of selection. In the case of \( f_1(x) = (x-1)^3 \), the coefficients \( \binom{3}{0} \), \( \binom{3}{1} \), \( \binom{3}{2} \) and \( \binom{3}{3} \) correspond to the numbers 1, 3, 3, and 1 respectively. These numbers determine how much weight each corresponding term in the polynomial has. Without these coefficients, our expansion would not accurately represent the original binomial expression.

Graphical Verification

Graphical verification is a visual method to ensure that our polynomial expansion is correct. After computing the expansion algebraically, we can use a graphing utility to plot the original function and its expanded form. If our expansion is accurate, the graphs of both the original function and its expansion will be indistinguishable from one another.

Putting It to the Test

For the function \( f_1(x) = (x-1)^3 \), we can plot the expanded form \( x^3 - 3x^2 + 3x - 1 \) and compare it with the graph of \( (x-1)^3 \). The graphs should overlay perfectly, confirming that our expansion mirrors the behavior of the original function across the entire domain. This not only provides a reassuring check on our work but also offers a tangible connection between abstract algebraic expressions and their geometric representations on a graph.