/*! 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 19 Give the leading term. $$ x^... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 19

Give the leading term. $$ x^{8} $$

Short Answer

Expert verified
Answer: The leading term of the polynomial $x^8$ is $x^8$.

Step by step solution

01

Identify the leading term

The given polynomial is: $$ x^8 $$ Since it's a monomial (only one term), the leading term is: $$ x^8 $$

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.

Monomial
A monomial is the simplest type of algebraic expression. It consists of only one term. This term can be a product of numbers, variables, and positive integer exponents.
For example, in the expression \(x^8\), the monomial is simply one variable raised to a power.
  • Monomials can have coefficients, like \(5x^3\).
  • Without a coefficient, it's assumed to be 1, as in \(x^8\) which is actually \(1x^8\).
  • The degree of a monomial is the sum of the exponents of all included variables. Here, \(x^8\) has a degree of 8.
Understanding monomials helps in building more complex algebraic expressions.
Polynomials
Polynomials are algebraic expressions that consist of multiple terms. Each term in a polynomial is a monomial. For example, \(3x^2 + 2x + 1\) is a polynomial with three terms.
Polynomials can be:
  • Monomials: Single term like \(x^8\).
  • Binomials: Two terms like \(x^2 + 3\).
  • Trinomials: Three terms like \(x^2 + 3x + 2\).
The leading term of a polynomial is the term with the highest degree. In a single term polynomial, such as \(x^8\), this term is easily identifiable as there is only one. For longer polynomials, listing terms in order of decreasing degree makes it easier to spot the leading term.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operators (like addition and subtraction). They represent mathematical relationships or quantities.
Some important features include:
  • They do not include an equals sign, differentiating them from equations.
  • Variables can take different values, making expressions flexible for various scenarios.
  • They can be as simple as a monomial, or more complex, like polynomials.
Understanding algebraic expressions involves learning to manipulate them using rules of arithmetic and algebra. This helps in simplifying and solving equations, as well as in expressing real-world problems mathematically.

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

State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The degree of \(p(x) q(x)\).

List the nonzero coefficients of the polynomials. $$ \frac{s^{13}}{3} $$

Find the constants \(r, s, p,\) and \(q\) if multiplying out the polynomial \(\left(r x^{5}+2 x^{4}+3\right)\left(2 x^{3}-s x^{2}+p\right)\) gives \(6 x^{8}-11 x^{7}-10 x^{6}-12 x^{5}-8 x^{4}+q x^{3}-15 x^{2}-12 .\)

Find possible formulas for the polynomials described. The degree is \(n=2\) and the zeros are \(x=2,-3\).

What is the coefficient of \(x^{n-1}\) in \((x+1)^{n}\) for \(n=2,3\) and \(4 ?\)
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Discrete Mathematics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Probability and Statistics

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.