Ch. 1 - Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 1 - Functions

Problem 82c

Chapter 1, Problem 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

Verified step by step guidance

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 $.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

Recommended video:

6:04

Introduction to Polynomial Functions

Related Practice

Textbook Question

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$ƒ (x) = x^{5} - x^{3} - 2$

328

views

Textbook Question

Symmetry Determine whether the graphs of the following equations and functions are symmetric about the x-axis, the y-axis, or the origin. Check your work by graphing.

$x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1$

311

views

Textbook Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$$\tan\]\left$($\tan$^{-1}1$\right$)$

256

views

Textbook Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$$\tan$^{-1}$\left$($\tan\[\left$($\frac{3\pi}{4}\]\right$)$\right$)$

258

views

Textbook Question

A culture of bacteria has a population of $150$ cells when it is first observed. The population doubles every $12~$\text{hr}$$ , which means its population is governed by the function $p$\left$(t$\right$)=150$\cdot{2^{\frac{t}{12}$}}$ , where $t$ is the number of hours after the first observation.