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Chapter 4: Problem 8

___________ is a monomial with coefficient 8 , in the variable \(x\), having degree 5 .

Short Answer

Expert verified
The monomial is 8x^5.

Step by step solution

01

Understand the Components of a Monomial

A monomial is an algebraic expression consisting of one term. It includes a coefficient (a numerical value) and variables raised to powers (exponents). In this exercise, the monomial needs a coefficient of 8, a variable of x, and a degree of 5.
02

Identify the Given Information

From the problem, the coefficient is 8, the variable is x, and the degree is 5.
03

Construct the Monomial

Combine the coefficient, variable, and degree. The coefficient is 8, the variable is x, and the degree (or exponent) is 5, so the monomial is written as: 8x^5

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Key Concepts

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

Coefficient
In algebra, a coefficient is the numerical part of a term. It is the number that stands in front of the variable in an algebraic expression.

For example, in the monomial \(8x^5\), the coefficient is 8. It's important because it tells us how many times the term is being counted.

Coefficients can be positive, negative, or even fractional. They make mathematics versatile and applicable to real-world scenarios. If you imagine 8 apples multiplied by some function of x, the coefficient (8) tells you how many groups of that function you are working with.
  • Positive coefficients contribute to positive values in functions.
  • Negative coefficients contribute to negative values, often implying direction or loss.
  • Fractional coefficients divide the term by a certain value.
Variable
A variable is a symbol used to represent an unknown value. In most cases, it is a letter from the alphabet, commonly 'x'.

Variables allow us to generalize mathematical problems. They hold places for potential values and enable algebraic manipulation. In the monomial \(8x^5\), 'x' is the variable. This 'x' could be any number, and it varies each time you solve the equation for a different value.

Variables are foundational in creating equations and functions. They give flexibility and help in forming relationships between numbers and operations.
  • Variables can represent any type of number.
  • They help in constructing expressions and equations.
  • In graphs, variables often correspond to axes, such as 'x' and 'y'.
Degree
The degree of a monomial is the highest exponent of the variable in that term. It tells us how many times the variable is multiplied by itself.

In the monomial \(8x^5\), the degree is 5. This means 'x' is raised to the power of 5, or \(x \times x \times x \times x \times x\). The degree is crucial because:

  • It indicates the term's growth rate. Higher degrees usually mean faster growth.
  • It determines the monomial's position in polynomial equations.
  • In calculus, the degree helps in deriving and integrating functions.
Understanding the degree gives insight into the behavior of the function or equation it represents. A higher degree can significantly change the outcome and interpretation of an expression.

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