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Chapter 5: Problem 10

Determine whether each expression is a polynomial. \(-2 x^{5}+9-7 x^{2}\)

Short Answer

Expert verified
Yes, \(-2 x^{5} + 9 - 7 x^{2}\)is a polynomial.

Step by step solution

01

Understand the Form of a Polynomial

A polynomial is an expression that can be written in the form: \[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]where \(a_n, a_{n-1}, \ldots, a_0\) are constants called coefficients, and \(n\) is a non-negative integer.
02

Identify the Terms in the Expression

The given expression is \(-2 x^{5} + 9 - 7 x^{2}\).Let's identify each term and its coefficient:- \(-2 x^{5}\) (coefficient: -2, exponent: 5)- \(-7 x^{2}\) (coefficient: -7, exponent: 2)- \(9\) (coefficient: 9, exponent: 0).
03

Check for Polynomial Requirements

To be a polynomial, each term must have a non-negative integer exponent:- For \(-2 x^{5}\), the exponent is 5 (non-negative integer).- For \(-7 x^{2}\), the exponent is 2 (non-negative integer).- For \(9\), the exponent is 0 because \(9 = 9 x^0\). (0 is a non-negative integer).
04

Conclusion

Since all terms have non-negative integer exponents and are in the form of a polynomial, the given expression \(-2 x^{5} + 9 - 7 x^{2}\)is indeed a polynomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Expression
A polynomial expression is a key concept in algebra. It is made up of terms, which can be constants, variables raised to non-negative integer exponents, or products of constants and variables. Each term in a polynomial is of the form \(a_n x^n\), where \(a_n\) is the coefficient and \(n\) is a non-negative integer exponent. In simple terms, think of a polynomial as a collection of these terms combined using addition or subtraction.

For example, the expression \(-2 x^5 + 9 - 7 x^2\) is a polynomial because it consists of three terms: \(-2 x^5\), \(9\), and \(-7 x^2\). All these terms follow the structure of polynomials. A polynomial can have one or more terms, and each term has a specific role in the expression.
Coefficients
Coefficients are the numerical factors in the terms of a polynomial. They are the constants that multiply the variables raised to an exponent in each term. In the polynomial expression \(-2 x^5 + 9 - 7 x^2\), the coefficients are -2, 9, and -7.

Let's break it down:
  • In the term \(-2 x^5\), -2 is the coefficient.
  • In the term \(9\), 9 is the coefficient. This term is often referred to as the constant term because it does not have a variable component.
  • In the term \(-7 x^2\), -7 is the coefficient.
Understanding coefficients is crucial because they determine the magnitude and direction (positive or negative) of the terms in a polynomial. They play a significant role in the behavior and shape of the polynomial's graph.
Non-negative Integer Exponents
Non-negative integer exponents are exponents that are whole numbers greater than or equal to zero. They are essential in defining a polynomial expression. For an expression to be considered a polynomial, each variable must be raised to a non-negative integer exponent.

In the polynomial \(-2 x^5 + 9 - 7 x^2\):
  • The exponent in \(-2 x^5\) is 5, which is a non-negative integer.
  • The exponent in \(9\) is implicitly 0, since \(9 = 9 x^0\). Zero is also a non-negative integer.
  • The exponent in \(-7 x^2\) is 2, which is a non-negative integer.
This strict requirement of non-negative integer exponents ensures that polynomial expressions have well-defined terms. It distinguishes polynomials from other types of mathematical expressions, like those involving negative exponents or fractional exponents.

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Most popular questions from this chapter

F(x)\( and \)g(x)\( are as given. Find a simplified expression for \)F(x)\( if \)F(x)=(f / g)(x)$. $$ f(x)=6 x^{2}-11 x-10, g(x)=3 x+2 $$

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Computer spreadsheet applications allow values for cells in a spreadsheet to be calculated from values in other cells. For example, if the cell Cl contains the formula $$ =\mathrm{A} 1+2 * \mathrm{B} 1 $$ the value in Cl will be the sum of the value in Al and twice the value in B1. This formula is a polynomial in the two variables Al and B1. The cell D6 contains the formula $$ =\mathrm{Al}-0.2 * \mathrm{B} 1+0.3^{*} \mathrm{Cl} $$ What is the value in \(\mathrm{D} 6\) if the value in \(\mathrm{Al}\) is \(10,\) the value in \(\mathrm{B} 1\) is \(-3,\) and the value in \(\mathrm{Cl}\) is \(30 ?\)

For each pair of functions fand \(g\), determine the domain of the sum, the difference, and the product of the two functions. $$ \begin{aligned} &f(x)=\frac{x}{2 x-9}\\\ &g(x)=\frac{5}{1-x} \end{aligned} $$
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