Problem 12 - Analysis 1: Ein Lehr- und Arbeitsbuch für Studienanfänger

Question

Berechnen sie mit Hilfe des Horner-Schemas die Funktionswerte \(p(2)\) und \(p(-2)\) für das polynom \(p(x)=x^{4}-5 x^{3}+5 x^{2}+5 x-6\)

Step-by-Step Solution

Answer

\(p(2) = 0\) and \(p(-2) = 60\).

Step 1: Set Up Horner's Scheme for p(2)

First, list the coefficients of the polynomial \(p(x) = x^4 - 5x^3 + 5x^2 + 5x - 6\) which are \([1, -5, 5, 5, -6]\). We will now calculate \(p(2)\) using Horner's method. Begin by writing these coefficients in a row.

Step 2: Apply Horner's Scheme for p(2)

Start with the first coefficient (1): 1. Multiply by 2 and add to the next coefficient (-5): \(1 \times 2 + (-5) = -3\).2. Multiply -3 by 2 and add to the next coefficient (5): \(-3 \times 2 + 5 = -1\).3. Multiply -1 by 2 and add to the next coefficient (5): \(-1 \times 2 + 5 = 3\).4. Multiply 3 by 2 and add to the last coefficient (-6): \(3 \times 2 + (-6) = 0\).The remainder is 0, which means \(p(2) = 0\).

Step 3: Set Up Horner's Scheme for p(-2)

Using the same coefficients \([1, -5, 5, 5, -6]\) for the polynomial \(p(x)\), we now calculate \(p(-2)\) with Horner's method. Begin by writing these coefficients in a row.

Step 4: Apply Horner's Scheme for p(-2)

Start with the first coefficient (1): 1. Multiply by -2 and add to the next coefficient (-5): \(1 \times (-2) + (-5) = -7\).2. Multiply -7 by -2 and add to the next coefficient (5): \(-7 \times (-2) + 5 = 19\).3. Multiply 19 by -2 and add to the next coefficient (5): \(19 \times (-2) + 5 = -33\).4. Multiply -33 by -2 and add to the last coefficient (-6): \(-33 \times (-2) + (-6) = 60\).The remainder is 60, which means \(p(-2) = 60\).

Key Concepts

Polynomial Evaluation

When working with polynomials, one of the essential tasks is evaluating them at given points. This means finding the value of the polynomial when replacing the variable with a specific number. For instance, in the polynomial \(p(x) = x^4 - 5x^3 + 5x^2 + 5x - 6\), we want to calculate \(p(2)\) and \(p(-2)\). Replacing \(x\) with these values in the polynomial gives us the function values, effectively evaluating the polynomial at these points.
However, manually substituting each value and computing can be tedious, especially for higher-degree polynomials. That's where efficient algorithms like Horner's method come into play, simplifying the process and reducing potential calculation errors.

Synthetic Division

Synthetic division is a streamlined method of dividing polynomials, useful particularly for simplifying polynomial evaluations like those required in Horner's method. It involves only the coefficients of the polynomial, making calculations much faster.
The key steps in synthetic division include:

• Aligning the coefficients of the polynomial in a row.
• Performing iterative multiplication and addition using the substitution value.
• Obtaining the final result through these quick calculations, which directly correspond to polynomial evaluations.

This process is efficient compared to traditional polynomial division and can also give insights into the zeros of polynomials. With practice, it becomes a powerful tool to solve polynomial-related problems.

Polynomial Roots

Roots of a polynomial are the values of \(x\) for which the polynomial equals zero. In practical terms, these are the points where the graph of the polynomial crosses the \(x\)-axis.
Finding roots is crucial because:

• It helps understand the behavior of the polynomial's graph.
• It is essential in solving polynomial equations.

Horner’s method not only aids in polynomial evaluation but also provides insight into roots. When the remainder is zero after applying Horner’s method with a specific value, it implies that value is a root of the polynomial. Thus, Horner's method serves a dual purpose of checking possible roots through polynomial evaluation.

Mathematical Algorithms

Mathematical algorithms like Horner's method transform complex mathematical problems into manageable steps. These algorithms often break calculations into smaller, repeatable operations, allowing for insights and solutions to emerge more elegantly.
Horner's method, for example, is an algorithm that is not just about evaluating polynomials but helps to uncover patterns and roots with minimal calculations, demonstrating the power of algorithmic thinking. It showcases how algorithmic solutions simplify and provide precision in solving mathematical problems.