/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 26 Use Lagrange interpolation to fi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 26

Use Lagrange interpolation to find the unique polynomial \(f(x)\) in \(\mathbb{R}[x]\) of the indicated degree \(n\) such that the graph of \(f(x)\) goes through the indicated points in the plane \(\mathbb{R}^{2}\). $$ n=2 \quad \text { points }(0,2)(1,0) \quad(2,0) $$

Short Answer

Expert verified
The unique polynomial is \(f(x) = x^2 - 3x + 2\).

Step by step solution

01

Identify the Interpolation Polynomial

The Lagrange interpolation polynomial for the given points \((0, 2), (1, 0), (2, 0)\) is a quadratic polynomial since \(n = 2\).
02

Define the Lagrange Basis Polynomials

The Lagrange basis polynomials \(L_i(x)\) are calculated by considering the other points except \(x_i\). For the given points, these are:- \(L_0(x) = \frac{(x-1)(x-2)}{(0-1)(0-2)} = \frac{(x-1)(x-2)}{2}\)- \(L_1(x) = \frac{(x-0)(x-2)}{(1-0)(1-2)} = -x(x-2)\)- \(L_2(x) = \frac{(x-0)(x-1)}{(2-0)(2-1)} = \frac{x(x-1)}{2}\)
03

Construct the Interpolation Polynomial

Using the Lagrange basis polynomials, the interpolation polynomial is constructed as:\[ f(x) = 2L_0(x) + 0L_1(x) + 0L_2(x) = 2 \cdot \frac{(x-1)(x-2)}{2} \] which simplifies to \[ (x-1)(x-2) \]
04

Simplify and Verify the Polynomial

Simplify the expression:\[ (x-1)(x-2) = x^2 - 3x + 2 \]Verify by checking the points:- For \(x=0\), \(f(0)=0^2 - 3 \cdot 0 + 2 = 2\), correct.- For \(x=1\), \(f(1)=1^2 - 3 \cdot 1 + 2 = 0\), correct.- For \(x=2\), \(f(2)=2^2 - 3 \cdot 2 + 2 = 0\), correct.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Polynomial Interpolation
Polynomial interpolation is a mathematical method used to estimate a polynomial that perfectly passes through a given set of data points. It's a valuable tool in numerical analysis and helps to approximate functions, making predictions based on discrete data. For any given set of data points, polynomial interpolation can determine a unique polynomial of the lowest degree that fits through all these points.

In the exercise, we use Lagrange interpolation to find a polynomial of degree two (quadratic polynomial) that fits the points (0, 2), (1, 0), and (2, 0). This is because three points require a quadratic polynomial, thus satisfying the condition for a unique solution.
  • The Lagrange interpolation method provides a systematic way to calculate this polynomial, ensuring that it is the simplest polynomial that fits the given data.
  • Polynomial interpolation is sensitive to the choice of points, meaning even a small change in one of the data points can lead to a completely different polynomial.
  • The resultant polynomial is an exact fit for the given points, ensuring that no error exists at these points.
Quadratic Polynomial
A quadratic polynomial is a type of polynomial characterized by the degree of two, which means its highest exponent on the variable is two. The general form of a quadratic polynomial is given by:\[ ax^2 + bx + c \]where \(a\), \(b\), and \(c\) are constants, and \(aeq 0\).

In this exercise, the quadratic polynomial derived was \(x^2 - 3x + 2\). This polynomial has the following attributes:
  • It has a highest degree of 2, fitting the requirement for a quadratic polynomial.
  • The coefficients, in this case, are derived based on fitting the polynomial to the specific points provided.
  • When evaluated at the given points, \((0, 2), (1, 0), (2, 0)\), the polynomial yields exactly the corresponding y-values, confirming its correctness.

Quadratic polynomials often appear in real-world applications such as physics, engineering, and economics, making the understanding of their construction important.
Lagrange Basis Polynomials
Lagrange basis polynomials (L_0(x), L_1(x), L_2(x)) are the building blocks used in creating the Lagrange interpolation polynomial. Each basis polynomial corresponds to one of the data points and is constructed in such a way that:
  • The polynomial is equal to 1 at its corresponding data point.
  • It is equal to 0 at all other data points.

For the given points, the Lagrange basis polynomials are calculated as:
  • L_0(x) = \frac{(x-1)(x-2)}{(0-1)(0-2)} = \frac{(x-1)(x-2)}{2}
  • L_1(x) = \frac{(x-0)(x-2)}{(1-0)(1-2)} = -x(x-2)
  • L_2(x) = \frac{(x-0)(x-1)}{(2-0)(2-1)} = \frac{x(x-1)}{2}

The final interpolation polynomial \(f(x)\) is constructed as a weighted sum of these basis polynomials, where the weight of each basis polynomial is the y-value of its corresponding data point. For instance, in the exercise, \(f(x)\) is simplified as:\[f(x) = 2L_0(x) + 0L_1(x) + 0L_2(x) = (x-1)(x-2)\]

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find all zeros of the indicated \(f(x)\) in the indicated field. $$ f(x)=x^{3}+3 x+5 \quad \text { in } \mathbb{Z}_{7} $$

Show that \(\mathbb{R}[x] /\left\langle x^{2}+1\right\rangle\) is isomorphic to \(C\).

Let \(f(x)\) and \(g(x)\) be two nonzero polynomials in \(F[x]\), where \(F\) is a field. Show that (a) \(\langle f(x)\rangle+\langle g(x)\rangle=\langle\operatorname{gcd}(f(x), g(x))\rangle\) (b) \(\langle f(x)\rangle \cap\langle g(x)\rangle=\langle\operatorname{lcm}(f(x), g(x))\rangle\)

Compute the sum \(f(x)+g(x)\) and product \(f(x) \cdot g(x)\) for the given polynomials \(f(x)\) and \(g(x)\) in the given polynomial ring \(R[x]\). $$ 6 x^{2}+2 \quad \text { and } 2 x^{3}+6 x \quad \text { in } \mathbb{Z}_{4}[x] $$

Determine whether the indicate quotient rings are fields. Justify your answers. $$ C[x] /\langle x-2\rangle $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.