Question

Consider the polynomial written in standard form: \(5 x^{4}+3 x^{3}+4 x^{2}+7 x-5 .\) a) Write the polynomial in its nested form. (See the previous problem.) b) How many multiplications does the nested form require when we evaluate the polynomial at a real number? How many multiplications does the standard form require? Can you generalize your answer to any \(n\) th degree polynomial?

Short answer

Answer: When using the nested form (Horner's method) to evaluate a polynomial, we save one multiplication compared to using the standard form. This is because the nested form requires n-1 multiplications for an nth degree polynomial, whereas the standard form requires n multiplications.

Step-by-step solution

Understanding the nested form of a polynomial

The nested form of a polynomial (Horner's method) represents a polynomial written in a way to minimize the number of multiplications required for evaluating the polynomial at a specific value of the variable. A general nth degree polynomial can be written as: \[P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\] In its nested form, it can be represented as: \[P(x) = (...(((a_n x) + a_{n-1})x + \ldots + a_1)x + a_0\]

Writing the polynomial in its nested form

The given polynomial in standard form is: \[P(x) = 5x^4 + 3x^3 + 4x^2 + 7x - 5\] Now, we will write it in its nested form (Horner's method): \[P(x) = ((((5x) + 3)x + 4)x + 7)x - 5\] Therefore, the nested form of the polynomial is: \[P(x) = ((((5x) + 3)x + 4)x + 7)x - 5\]

Counting the number of multiplications for both forms

Standard form of the polynomial: \[P(x) = 5x^4 + 3x^3 + 4x^2 + 7x - 5\] There are 4 multiplications (\(x\) raised to 4, 3, 2, and 1) required in the standard form. Nested form (Horner's method) of the polynomial: \[P(x) = ((((5x) + 3)x + 4)x + 7)x - 5\] There are 3 multiplications (\(5x\), the result by \(x\), and the result by \(x\)) required in the nested form.

Generalizing the number of multiplications for any nth degree polynomial

For any nth degree polynomial in standard form, there will always be n multiplications required (since we have terms from \(x^n\) to \(x^1\)). However, for any nth degree polynomial in the nested form (Horner's method), there will always be n-1 multiplications required (since we start with the coefficient of \(x^n\) and then multiply the result with \(x\) up to \((n-1)\) times). This is because after the first multiplication, each subsequent multiplication incorporates the previous result and saves one multiplication. Thus, using Horner's method can save computationally expensive multiplications for large polynomials.

Key concepts

Horner's Method

Horner's method is a highly efficient way to evaluate polynomials, and it's especially useful for computations by hand or within computer algorithms due to its ability to reduce the number of required multiplications.

Imagine a polynomial like a nest of boxes, with each box containing a smaller one. Starting with the largest exponent term, you unwrap each layer one by one until you reach the constant term. This 'nested' process transforms a polynomial into a less complex form that is quicker to evaluate.

For instance, the polynomial \(5x^4 + 3x^3 + 4x^2 + 7x - 5\) becomes \( ((((5x) + 3)x + 4)x + 7)x - 5\) when applying Horner's method. Notice how it starts with the highest degree term and packs the polynomial into a more compact form. This approach significantly impacts computational speed, particularly when dealing with high-degree polynomials.

Polynomial Evaluation

Evaluating a polynomial means calculating its value for a specific input, usually denoted as \(x\). The traditional method involves replacing \(x\) with the given number in all terms and performing the arithmetic operations accordingly. This can be a relentless task if done frequently, or with polynomials of a high degree.

Using Horner's method for polynomial evaluation changes the game, as it allows a step-by-step simplification that reduces workload. Like unboxing layers until you reach the core, it progressively simplifies the polynomial. Through its systematic approach, Horner's method not only streamlines the process but also minimizes potential errors that might occur when dealing with multiple terms and operations.

Multiplications in Polynomials

Multiplications in polynomials often dictate the complexity of polynomial computations. More multiplications typically mean more work and higher chances of miscalculation.

The standard polynomial form demands a number of multiplications equal to the polynomial degree. For example, a cubic polynomial requires three multiplications: \(x^3\), \(x^2\), and \(x^1\).

Horner's method shines by reducing the multiplication count to one less than the polynomial's degree. The nth degree polynomial \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\) in standard form requires \(n\) multiplications, while its nested form using Horner's method requires only \(n-1\). This reduction makes a substantial difference in computational effort, especially when dealing with higher-degree polynomials.

Nth Degree Polynomial

An nth degree polynomial is a polynomial whose highest exponent on the variable \(x\) is \(n\). These polynomials can become quite complex as their degree increases, presenting challenges in their manipulation and evaluation.

In standard form, an nth degree polynomial is written as \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\), with each term contributing to the overall shape and behavior of the polynomial graph. But when it comes to computations, the degree tells us how many multiplications we're in for.

In practice, the difference between evaluating an nth degree polynomial in standard form versus using Horner's method is evident; the standard form requires \(n\) multiplications, while Horner's method reduces it to \(n-1\). Thus, understanding and utilizing the nested form can be a powerful tool for anyone working with polynomials, making calculations faster and more efficient.