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Chapter 1: Problem 50

For \(n\) a positive integer, find the smallest positive integer \(d=d(n)\) for which there exists a polynomial of degree \(d\) whose graph passes through the points \((1,2),(2,3), \ldots,(n, n+1)\), and \((n+1,1)\) in the plane.

Short Answer

Expert verified
The smallest positive integer \( d = n \).

Step by step solution

01

Identify Given Points and Degrees of Polynomial

The points given are \( (1,2), (2,3), \ldots, (n, n+1), (n+1,1) \). To pass through these \( n+1 \) points using a polynomial of degree \( d \), the polynomial must have at least \( n+1 \) coefficients. Thus, the degree \( d \) should be at least \( n \).
02

Determine the Smallest Polynomial Degree

To ensure the polynomial degree \( d \) fits all \( n+1 \) points exactly, \( d \) must be \( d = n \). Any lower degree would not provide enough flexibility to fit these points. Thus, \( d = n \) is the smallest needed degree.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

degree of polynomial
In mathematics, the degree of a polynomial is the highest power of the variable in the polynomial. For instance, in the polynomial equation \( ax^n + bx^{n-1} + \ldots + c \), the degree is \( n \). If you have a set of points, the smallest degree of polynomial that can pass through these points is one less than the number of points. For our exercise, we need a polynomial to pass through \( n+1 \) points. Hence, the degree of this polynomial must be at least \( n \). This ensures that we have enough 'room' within the polynomial's terms to match each point given. When you graph this polynomial, it will have a curve that changes direction up to \( n \) times.
graph theory
Graph theory is an area of mathematics that studies the relationships and connections between nodes. These nodes are often called 'vertices' and the connections 'edges'. In our problem, the given points \( (1,2), (2,3), \ldots, (n, n+1), (n+1,1) \) can be seen as nodes in the graph. The edges would be the polynomial curve that connects these nodes. The task of finding a polynomial that intersects these points is comparable to finding a path in graph theory that visits each node precisely once. This idea is akin to constructing graphs where each vertex (point) needs to be covered.
coordinate geometry
Coordinate geometry, also known as analytic geometry, uses coordinates and algebra to study geometric problems. In our exercise, we are given specific coordinates \( (1,2), (2,3), \ldots, (n, n+1), (n+1,1) \). These points lie in the two-dimensional coordinate plane, and we need to find a polynomial that passes through all of them. The polynomial's graph represents a smooth curve that intersects the x-y plane at these points. Understanding how these points are situated helps visualize how the polynomial must curve and bend to meet the requirement.
polynomial functions
Polynomial functions are expressions like \( f(x) = ax^n + bx^{n-1} + \ldots + c \) where powers of \( x \) are whole numbers and coefficients \( a, b, c \) are constants. The goal in our exercise is to determine a polynomial that fits through given points. A polynomial function can be tailored to fit a set of points by adjusting its coefficients. For a polynomial of degree \( n \), it can be uniquely determined by \( n+1 \) points. In our problem, we deduce the degree of the polynomial needs to match the number of points minus one. Thus, to fit the points \( (1,2), (2,3), \ldots, (n, n+1), (n+1,1) \), which are \( n+1 \) points, a polynomial of degree \( n \) is necessary.

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Most popular questions from this chapter

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