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Chapter 5: Problem 11

Give the numerical coefficient and the degree of each term. $$ -x^{3} $$

Short Answer

Expert verified
Coefficient: -1, Degree: 3.

Step by step solution

01

Identify the term

The given term is \[ -x^{3} \] which is a polynomial with one single term.
02

Find the numerical coefficient

To find the numerical coefficient, look at the number in front of the variable. Here, \[ -x^{3} \] is the same as \[ -1 \times x^{3} \], so the numerical coefficient is \[ -1 \].
03

Determine the degree of the term

The degree of a term is the exponent of the variable. For \[ -x^{3} \], the exponent is \[ 3 \], so the degree is \[ 3 \].

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Key Concepts

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

numerical coefficient
When working with polynomials, one important concept is the numerical coefficient. This is the number that multiplies the variable in a term. For example, in the term \( -x^{3} \), we can write it as \(-1 \times x^{3} \). Here, \-1\ is the numerical coefficient. The numerical coefficient can be a positive or negative number. If there is no visible number, it is assumed to be \(+1\) or \(-1\) depending on the sign.
degree of a term
The degree of a term in a polynomial is another key concept. It tells us the highest exponent the variable has in that term. For instance, in \( -x^{3} \), the variable is \( x \) and its exponent is \ 3 \. Therefore, the degree of the term \( -x^{3} \) is \ 3 \. Degrees are important because they help us understand the behavior and characteristics of the polynomial, especially when comparing different polynomials.
polynomial terms
Understanding polynomial terms is essential when studying polynomials. A polynomial can be made up of one or more terms, where each term is a product of a numerical coefficient and a variable raised to an exponent. For example, in the polynomial \( -x^{3} \), there is only one term: \( -x^{3} \). Each term in a polynomial has its own numerical coefficient and degree. Knowing how to identify and work with each term helps simplify and solve polynomial equations.

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