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Divide $ R(x) $ by $ x - 1 $ using synthetic division:

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Exploring the Hidden Significance of 1, 0, -3, 2: A Mathematical Journey Numbers may seem simple, but beneath their surface lies a fascinating world of meaning, logic, and application. The combination 1, 0, -3, 2—or more precisely, the integers 1, 0, -3, and 2—holds subtle but powerful significance across mathematics, science, and everyday problem-solving. In this article, we unpack the meaning, patterns, and real-world relevance behind these four numbers. --- ### Understanding the Numbers: 1, 0, -3, 2 At first glance, 1, 0, -3, and 2 are just whole numbers. But collectively, they represent key concepts in arithmetic, algebra, and data representation.

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$& & 1 & 1 & -2 \\$

& & 1 & 1 & -2 \\

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$& 1 & 1 & -2 & 0 \\$

& 1 & 1 & -2 & 0 \\

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So $ R(x) = (x - 1)(x^2 + x - 2) $. Factor the quadratic:

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x^2 + x - 2 = (x + 2)(x - 1).

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R(x) = (x - 1)^2(x + 2).

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The roots are $ x = 1 $ (with multiplicity 2) and $ x = -2 $. Therefore, there are two distinct real roots, but three real roots counting multiplicity. Since the question asks for the number of real roots (not distinct), the answer is:

Understanding the Real Roots of the Polynomial: Analyzing $ x = 1 $ (Multiplicity 2) and $ x = -2 $ When solving polynomial equations, understanding both the number of distinct real roots and the total real roots counting multiplicity is essential for accurate interpretation. In the case of the polynomial with roots $ x = 1 $ (with multiplicity 2) and $ x = -2 $, let’s break down how these roots shape the overall structure of the equation and its real solutions. ### What Does Multiplicity Mean? Multiplicity refers to the number of times a particular root appears in a polynomial. A root with multiplicity 2 (or higher) means that the graph of the polynomial touches but does not cross the x-axis at that point—instead, it “bounces” off. A simple root (multiplicity 1) results in a clear crossing of the x-axis. Here, the root $ x = 1 $ has multiplicity 2, while $ x = -2 $ appears once.

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$Question: A zoologist studying social behavior models the interaction between two animals with the function $ I(u) = u - \frac{u^3}{9} $. If $ n $ is a positive integer, define $ x_n $ as the value of $ u $ such that $ I(x_n) = \frac{1}{n} $. Find $ \lim_{n \to \infty} x_n $.$

Question: A zoologist studying social behavior models the interaction between two animals with the function $ I(u) = u - \frac{u^3}{9} $. If $ n $ is a positive integer, define $ x_n $ as the value of $ u $ such that $ I(x_n) = \frac{1}{n} $. Find $ \lim_{n \to \infty} x_n $.

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Divide $ R(x) $ by $ x - 1 $ using synthetic division:

\begin{array}{r|rrrr}

& 1 & 0 & -3 & 2 \\

& & 1 & 1 & -2 \\

& 1 & 1 & -2 & 0 \\

So $ R(x) = (x - 1)(x^2 + x - 2) $. Factor the quadratic:

x^2 + x - 2 = (x + 2)(x - 1).

R(x) = (x - 1)^2(x + 2).

The roots are $ x = 1 $ (with multiplicity 2) and $ x = -2 $. Therefore, there are two distinct real roots, but three real roots counting multiplicity. Since the question asks for the number of real roots (not distinct), the answer is:

Question: A zoologist studying social behavior models the interaction between two animals with the function $ I(u) = u - \frac{u^3}{9} $. If $ n $ is a positive integer, define $ x_n $ as the value of $ u $ such that $ I(x_n) = \frac{1}{n} $. Find $ \lim_{n \to \infty} x_n $.