Problem 72 - Algebra and Trigonometry

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 68 and 69 to verify the expansion. $$ f_{1}(x)=(x+2)^{6} $$

Step-by-Step Solution

Answer

The expanded form of function \(f_1(x) = (x+2)^6\) using Binomial theorem is \(f_1(x) = x^6 + 12x^5 +60x^4 + 160x^3 + 240x^2 + 192x +64\). This can be verified by comparing the graphs of the original and expanded functions, which should coincide with each other.

Step 1: Apply the Binomial Theorem

The correct application of the Binomial theorem can be used to expand the expression. The binomial theorem is given by: \[(a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^k\] Applying this theorem to \((x+2)^6\) gives us the expanded form: \[(x+2)^6 = {6 \choose 0} x^6(2)^0 + {6 \choose 1} x^5(2)^1 + {6 \choose 2} x^4(2)^2 + {6 \choose 3} x^3(2)^3 + {6 \choose 4} x^2(2)^4 + {6 \choose 5} x(2)^5 + {6 \choose 6} (2)^6\]

Step 2: Calculate the coefficients

Calculating the coefficients, the expanded form of the function is: \(f_1(x) = x^6 + 12x^5 +60x^4 + 160x^3 + 240x^2 + 192x +64\).

Step 3: Graph the function and it's expansion

Now graph out the function \(f_1(x) = (x+2)^6\) and \(f_1(x) = x^6 + 12x^5 +60x^4 + 160x^3 + 240x^2 + 192x +64\) using a graphing utility. If the Binomial theorem was applied correctly and calculations for coefficients are correct, both graphs will coincide with each other.

Key Concepts

Polynomial Expansion

Understanding the process of expanding a binomial expression into a polynomial is crucial for working with algebraic equations. The term 'polynomial expansion' refers to the process of taking a binomial raised to a power and expressing it as a polynomial, which is a sum of terms consisting of coefficients and variables raised to non-negative integer powers.

Let's take the expression \( (x+2)^6 \) as our example and apply the Binomial Theorem to carry out the expansion. This theorem serves as a formula to expand binomials raised to any positive integer exponent. Employing this theorem allows one to transform a compact binomial expression into an extended polynomial, giving detailed insight into the function's behavior.

When applying the Binomial Theorem, we generate terms by multiplying the binomial coefficients, which are specific numbers at each term of the expansion, with the appropriate powers of the terms in the original binomial. The result in our case is a sixth-degree polynomial. Each term of this polynomial reflects the possible combinations of \(x+2\) when multiplied by itself six times.

Graphing Utility

A graphing utility is an instructional technology that plays an essential role in visualizing mathematical concepts, including polynomial expansions. It takes equations, like our expanded \( f_1(x) = x^6 + 12x^5 +60x^4 + 160x^3 + 240x^2 + 192x +64 \), and translates them into graphical representations.

By entering both the original binomial expression and its expanded polynomial form into a graphing utility, students can directly observe that both expressions describe the same function. It's visual proof that reinforces their understanding of the Binomial Theorem and polynomial expansion. Moreover, when the graph of the original function \( (x+2)^6 \) is superimposed with the graph of its expanded form, the congruence of the two graphs serves as a verification step for the accuracy of the expansion obtained.

Binomial Coefficients

Binomial coefficients are the numbers that appear as the coefficients in the binomial expansion and they hold significant combinatorial meaning. They are denoted by \( {n \choose k} \), which represents the number of ways to choose \( k \) elements out of a total of \( n \) elements, without considering the order of selection.

In the context of our example, the binomial coefficients are the multipliers that determine the influence of each term in the polynomial after the expansion. For instance, in the expression \( {6 \choose 2} x^4(2)^2 \), the binomial coefficient \( {6 \choose 2}\) is calculated as \(\frac{6!}{2!(6-2)!}\), which equals 15. Here, \(6!\) denotes factorial of 6, a product of all positive integers up to 6. Thus, understanding these coefficients not only allows the student to carry out the expansion correctly but also provides insight into the combinatorial foundations of algebra.