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Chapter 5: Problem 4

Find the degree of each term. \(-z^{3}\)

Short Answer

Expert verified
The degree of \(-z^{3}\) is 3.

Step by step solution

01

Identify the Terms

The expression is given as \(-z^{3}\). Each term in this expression is separated by a plus (+) or minus (鈭�) sign. Since there is only one term here, which is \(-z^{3}\), we move on to the next step.
02

Review the Structure

In the term \(-z^{3}\), the expression consists of a variable \(z\) raised to the power of 3. This means that the only thing affecting the term's degree is this exponent.
03

Determine the Degree

The degree of a term involving a variable is determined by the exponent of that variable. Here, \(z\) is raised to the power of 3. Therefore, the degree of the term \(-z^{3}\) is 3.

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Key Concepts

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

Degree of a Term
The degree of a term is an important concept in understanding polynomials. It refers to the highest power (exponent) of the variable within that term. Consider the term \(-z^3\). Here, the variable is \(z\), and it is raised to the power of 3.
This means the degree of the term is 3. The degree helps tell us how "big" the term is in a polynomial and is used to compare terms within the same polynomial.
For single terms, finding the degree is straightforward: just check the exponent.
  • If you have other terms like \(x^5\) or \(w^7\), their degrees would be 5 and 7 respectively.
  • In a polynomial with multiple terms, the degree of the polynomial is the highest degree of any single term.
Recognizing the degree of each term is crucial for understanding the overall structure and properties of polynomials.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations like addition, subtraction, multiplication, and division.
These expressions can be as simple as a single number or a variable, or much more complex, like \(3x^2 + 2x - 5\).
They form the foundation for equations and functions in algebra. In the expression \(-z^3\), we have:
  • A single term with a variable \(z\) and an operation (raising to the power).
  • The negative sign in front of \(z^3\) is also part of this expression.
Understanding algebraic expressions helps break down complex problems into simpler parts.
They are the building blocks used in solving equations, graphing lines, and modeling real-world scenarios.
Exponents
Exponents are a shorthand way to express repeated multiplication of a number by itself.
In the term \(-z^3\), the exponent is the 3, indicating that \(z\) is multiplied by itself three times. Exponents have key rules that make calculations easier:
  • **Zero Exponent Rule:** Any number raised to the power of zero equals 1, \(a^0 = 1\).
  • **Product of Powers Rule:** Multiply numbers and add their exponents, \(a^m \times a^n = a^{m+n}\).
  • **Power of a Power Rule:** Multiply the exponents, \((a^m)^n = a^{m\times n}\).
Getting comfortable with exponents will help in evaluating expressions and solving problems in algebra.
They indicate not only power and magnitude but also play a crucial role in calculus and higher mathematics.

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Most popular questions from this chapter

Use substitution to factor each polynomial completely. See Examples 11 and 12. $$ (3 x-1)^{2}+5(3 x-1)+6 $$

Without calculating, determine which number is larger. $$ 7^{-11} \text { or } 7^{-13} $$

Factor each polynomial completely. See Examples 1 through 12. $$ 3 x^{2} y^{3}+6 x^{2} y^{2}-45 x^{2} y $$

Multiply. See Section 5.4. $$ (2 x+1)^{2} $$

Find the value of \(c\) that makes each trinomial a perfect square trinomial. $$ m^{2}-14 m+c $$
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