Problem 68 - 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. $$ f_{1}(x)=(x+2)^{6} $$

Step-by-Step Solution

Answer

The polynomial expansion for the function \(f_{1}(x) = (x+2)^{6}\) is \(f_{1}(x) = x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64\). This has been verified using a graphing utility by confirming the overlap of the graph of the original function and the expanded polynomial.

Step 1: Understand the Binomial Theorem

The Binomial Theorem states that \( (x+y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k\). This theorem will be used to expand the given binomial.

Step 2: Apply the Binomial Theorem to expand \(f_{1}(x)\)

For \(f_{1}(x) = (x+2)^{6}\), apply the Binomial Theorem to get the expanded form: \(f_{1}(x) = \sum_{k=0}^{6} {6 \choose k} x^{6-k} 2^k\). Directly applying this we get: \(f_{1}(x) = {6 \choose 0} x^6 2^0 + {6 \choose 1} x^5 2^1 + {6 \choose 2} x^4 2^2 + {6 \choose 3} x^3 2^3 + {6 \choose 4} x^2 2^4 + {6 \choose 5} x^1 2^5 + {6 \choose 6} x^0 2^6\). This simplifies to \(f_{1}(x) = x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64\).

Step 3: Verification Using Graphing Utility

It is necessary to verify the expansion by using a graphing utility. Two graphs are obtained, one for \(f_{1}(x) = (x+2)^{6}\) and one for its expanded version \(f_{1}(x) = x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64\). If the expansion is correct, the two graphs coincide on all points. Thus verifying the obtained polynomial expansion.

Key Concepts

Polynomial Expansion

Polynomial expansion involves breaking down a binomial expression raised to a power into a sum of terms with integer coefficients. The Binomial Theorem is an essential tool in this process. It provides a formula to expand any expression in the form \((x + y)^n\). The theorem states: \[ (x + y)^n = \sum_{k=0}^{n} {n \choose k} x^{n-k} y^k \] Here, \({n \choose k}\) represents the binomial coefficients that determine the weights of each expanded term. This formula allows us to express a binomial raised to a power as a polynomial with multiple terms.
For example, to expand \((x + 2)^6\), apply the theorem to get: - \(x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64\). This expanded polynomial represents the original binomial in a different form, with each term having different powers of \(x\) and numeric coefficients.
Mastering polynomial expansion is crucial for simplifying higher power expressions and is widely used in algebra, calculus, and beyond.

Verification Using Graphing Utility

Verification using a graphing utility is a powerful way to confirm the accuracy of a polynomial expansion. It involves plotting graphs of the original expression and its expanded form. Both graphs are then compared to ensure they overlap completely, confirming the expansion is correct.
Here's how to do it: - Use a graphing calculator or software to plot the function \(f_{1}(x) = (x + 2)^6\). - Plot the expanded form \(f_{1}(x) = x^6 + 12x^5 + 60x^4 + 160x^3 + 240x^2 + 192x + 64\) on the same axes.
If the expansion is correct, the two graphs will coincide perfectly. This visual comparison is very effective, especially for complex expansions, as it immediately shows any discrepancies that might exist between the original and expanded forms.
This technique is not only useful for verifying expansions but also for understanding the behavior and characteristics of functions presented in different formats.

Combinatorial Coefficients

Combinatorial coefficients, often referred to as binomial coefficients, are crucial to the Binomial Theorem. They are denoted as \({n \choose k}\) and are calculated using the formula: \[ {n \choose k} = \frac{n!}{k!(n-k)!} \] These coefficients are the numbers that appear in Pascal’s Triangle and dictate how each term in a binomial expansion is weighted.
For example, in the expansion of \((x + 2)^6\), the coefficients are:

• \({6 \choose 0} = 1\)
• \({6 \choose 1} = 6\)
• \({6 \choose 2} = 15\)
• \({6 \choose 3} = 20\)
• \({6 \choose 4} = 15\)
• \({6 \choose 5} = 6\)
• \({6 \choose 6} = 1\)

These numbers help determine each term's contribution in the polynomial expansion, ensuring the correct construction of the expanded form.
Understanding combinatorial coefficients opens doors to solving a variety of problems in algebra and statistics, enabling more efficient computation and deeper insight into the structure of mathematical expressions.