Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 1 - Functions

Problema 82c

Capitolo 1, Problema 82c

Composition of polynomials
Let ƒ be an nth-degree polynomial and let g be an mth-degree polynomial.
What is the degree of the following polynomials?
ƒ ⋅ f

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Identify the degree of the polynomial $ f $, which is $ n $.

Understand that $ f \cdot f $ represents the composition of the polynomial $ f $ with itself.

Recall that the degree of a polynomial product $ f \cdot g $ is the sum of the degrees of $ f $ and $ g $.

Since $ f \cdot f $ is $ f $ composed with itself, the degree is $ n + n $.

Conclude that the degree of $ f \cdot f $ is $ 2n $.

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Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. For example, in the polynomial f(x) = 3x^4 + 2x^2 + 1, the degree is 4. The degree provides important information about the polynomial's behavior, including the number of roots and the end behavior of its graph.

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Multiplication of Polynomials

When multiplying polynomials, the degree of the resulting polynomial is the sum of the degrees of the polynomials being multiplied. For instance, if f is an nth-degree polynomial and g is an mth-degree polynomial, then the degree of the product f ⋅ g is n + m. This principle is crucial for determining the degree of polynomial expressions resulting from operations.

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Introduction to Polynomial Functions

Composition of Polynomials

The composition of polynomials involves substituting one polynomial into another. For example, if f(x) is a polynomial and g(x) is another, then the composition f(g(x)) results in a new polynomial. While the question focuses on multiplication, understanding composition helps clarify how polynomials interact, especially in more complex expressions.

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