Problem 57 - Contemporary Precalculus

Question

True or False. In Exercises \(56-58,\) use the technique of Exercise 55 to determine graphically whether the given state. ment is possibly true or definitely false. ( We say "possibly true " because nwo graphs that appear identical on a calculator screen may actually differ by small amounts or at places not shown in the window.) $$(1-x)^{6}=1-6 x+15 x^{2}-20 x^{3}+15 x^{4}-6 x^{5}+x^{6}$$

Step-by-Step Solution

Answer

Short Answer: After graphing the difference expression and analyzing the graph, if the graph appears to be the zero function (a horizontal line at y=0), then the statement is possibly true, as this would imply both sides of the equation are equal. If the graph is not the zero function, the statement is definitely false.

Step 1: Rewrite the given equation in the form of a difference

In order to graph the equation and compare the two sides, we need to rewrite the equation into a difference. We can do this by moving all the terms from the right side to the left side, that is: $$(1-x)^{6}-(1-6 x+15 x^{2}-20 x^{3}+15 x^{4}-6 x^{5}+x^{6})$$

Step 2: Graph the difference

Now we will graph the difference expression and observe the resulting graph. Since this is a high degree polynomial, it may be difficult to observe the graph in detail just by plotting it.

Step 3: Observe the graph

Upon graphing the difference expression, we will see if the graph appears to be the zero function (a horizontal line at y=0) or not. If the graph appears to be the zero function, then the given statement is possibly true, because the difference between the two sides of the equation is zero, implying they are equal. If the graph is not the zero function, then the statement is definitely false.

Step 4: State the conclusion

After analyzing the graph, state whether the given statement is possibly true or definitely false based on the observation of the difference graph. If the graph appears to be the zero function, the statement is possibly true. If it's not, the statement is definitely false.

Key Concepts

Polynomial Identity

Understanding the concept of a polynomial identity is essential for students grappling with algebra. Essentially, a polynomial identity is an equality that holds for all values of the variable involved. Think of it as a mask that, no matter who wears it, looks the same on everyone. Likewise, no matter which value we substitute into an identity, the expression retains its truth. An excellent example of a polynomial identity is the square of a binomial, such as \( (a+b)^2 = a^2 + 2ab + b^2 \). No matter what values you substitute for 'a' and 'b', this equation remains steadfast in its veracity.

In the context of graphing polynomial functions, utilizing these identities can greatly simplify the graphing process. By recognizing a complex expression as an identity, you can deduce characteristics of its graph without plotting every single point. The exercise provided, involving the expansion of \( (1-x)^{6} \) and its comparison to a polynomial string, demonstrates an application of polynomial identities. By juxtaposing the two expressions, we aim to identify if they are, indeed, the same polynomial, represented differently—an identity.

Graphical Verification

Graphical verification is like playing a game of spot the difference, but with functions. When we graph two expressions and their plots look indistinguishable, we move towards concluding that the expressions represent the same function or identity. However, there's always a slight chance of deception; just because two graphs look alike, doesn’t guarantee they are identical — they might differ subtly in ways not captured within our graph window. This is particularly true when using graphing calculators with limited resolution.

For students, it's crucial to learn how to properly set the window size and resolution on their graphing tool to detect these subtle differences. When the difference of two functions is graphed, as demonstrated in the exercise, we should ideally see a flat line along the x-axis if they are identical. If there's any deviation from this zero function, we've uncovered a divergence and hence, a false identity. It’s this intersection of visual confirmation and algebraic manipulation that gives students a robust understanding of polynomial functions and their behavior.

Binomial Theorem

Remember opening a treasure chest? The binomial theorem is like that for algebra—it unlocks the riches of expanding expressions raised to any power effortlessly. This theorem provides a formula for expanding binomials to any given power in terms of summation notation involving coefficients known as binomial coefficients. These coefficients can be found in Pascal's Triangle or calculated using combination formulae.

The generalized form of the binomial theorem is \( (a + b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k} b^{k} \), where \( {n \choose k} \) is the binomial coefficient, representing the number of ways to choose 'k' elements from a set of 'n' elements. In the context of the given exercise, the binomial theorem is applied to expand \( (1-x)^{6} \). Each term in the expansion corresponds to a term in Pascal's Triangle, and careful calculation reveals the polynomial in its expanded form. This not only helps in verifying the polynomial identity but also bolsters the understanding of the symmetry and patterns within polynomial expansions. Encouraging students to practice expanding binomials through this theorem is a great way to reinforce this method and instill confidence in recognizing polynomial identities.