/*! 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 55 The expression \(\frac{4 x^{3}}{... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 55

The expression \(\frac{4 x^{3}}{9 x-7}\) is not a polynomial because

Short Answer

Expert verified
Answer: The expression \(\frac{4x^3}{9x-7}\) is not a polynomial because it contains a variable in the denominator with a negative exponent. A polynomial is defined as an algebraic expression with variables, coefficients, and non-negative whole number exponents. In this case, the expression can be rewritten as \(4x^3(9x-7)^{-1}\), which contains a term with a negative exponent, thus not fitting the definition of a polynomial.

Step by step solution

01

Defining a Polynomial

A polynomial is an algebraic expression made up of variables, coefficients, and exponents, where exponents are non-negative whole numbers (integers greater than or equal to 0). A polynomial has the general form: \[P(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0\] Where \(a_n, a_{n-1},..., a_1, a_0\) are constants, and \(n\) is a non-negative whole number.
02

Identifying the Key Differences

In the given expression, \(\frac{4x^3}{9x-7}\), the numerator is a polynomial, \(4x^3\), as it meets the definition mentioned above. However, the denominator is also an algebraic expression, \(9x - 7\). This fraction doesn't fit the definition of a polynomial as it contains a variable in the denominator.
03

Explaining Why a Polynomial Cannot Have Fractions

A polynomial cannot have fractions because of the requirement that exponents must be non-negative whole numbers. If there is a fraction in the polynomial, it means there is a variable in the denominator with a negative exponent. In the case of the given expression, it can be rewritten as: \[\frac{4x^3}{9x-7} = 4x^3(9x-7)^{-1}\] As we can see, now there is a term with a negative exponent, which doesn't fit the definition of a polynomial. Therefore, the given expression is not a polynomial.

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.

Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They can represent a wide range of mathematical scenarios. Typically, they don't have an equals sign unless they're part of an equation.

An algebraic expression can be as simple as a number or consist of complex combinations:
  • Simple: Just a constant or a single variable like x or 7.
  • Complex: Can include multiple terms, such as 3x + 2y - 5.
In a fraction, the numerator or denominator can each be algebraic expressions. However, for a whole expression to be called a polynomial, it must not involve any variables in the denominator.
Exponents
Exponents represent repeated multiplication of the same number. In the expression x3, the exponent 3 tells us to multiply x by itself three times, or x 路 x 路 x.

Key characteristics of exponents in polynomials include:
  • Non-negative integers: All exponents in polynomials must be whole numbers (0, 1, 2,...).
  • Indicate power: They express how many times the variable is used in multiplication.
When a polynomial expression is turned into a fraction, like in \(\frac{4x^3}{9x-7}\), it might introduce negative exponents upon manipulation, which violate polynomial rules.
Variables and Coefficients
Variables are symbols used to represent unknown or changeable values, like x or y in algebraic expressions. They serve as placeholders that can take on many values.

Coefficients are the numbers in front of these variables, indicating their multiplicative factor.
  • In the term 4x3, 4 is the coefficient, and x is the variable.
  • Each term may consist of a coefficient multiplied by a variable raised to a power.
When forming polynomials, every term follows this structure without any terms having negative coefficients or exponents in the polynomial context.

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

Simplify the algebraic expressions for the following problems. $$ (3 a-7)(2 a+10) $$

For the following problems, simplify each of the algebraic expressions. $$ 9 a^{3} b^{7}\left(a^{3} b^{5}-2 a^{2} b^{2}+6\right)-2 a\left(a^{2} b^{7}-5 a^{5} b^{12}+3 a^{4} b^{9}\right)-a^{3} b^{7} $$

List, if any should appear, the common factors for the following problems. $$ 4(a+2 b)+6(a+2 b) $$

For the following problems, perform the multiplications and combine any like terms. $$ x^{3} y^{2}\left(5 x^{2} y^{2}-3\right)(2 x y-1) $$

For the following problems, find the products. $$ (2 h-8 k)^{2} $$
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

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.