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Chapter 3: Problem 14

Identify the coefficient and degree of each monomial. $$5 t^{2}$$

Short Answer

Expert verified
Answer: The coefficient of the monomial \(5t^2\) is 5, and its degree is 2.

Step by step solution

01

Identifying the Coefficient

In the given monomial \(5t^2\), the coefficient is the number that multiplies the variable, which in this case is 5. So, the coefficient of the monomial is 5.
02

Identifying the Degree

In the given monomial \(5t^2\), the degree is the exponent of the variable, which determines the highest power at which the variable is raised. For this monomial, the degree is 2, since the variable t is raised to the power of 2 (\(t^2\)). Final Answer: The coefficient of the monomial \(5t^2\) is 5, and its degree is 2.

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Key Concepts

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

Understanding the Coefficient
A monomial is a mathematical expression with one term. In a monomial like \(5t^2\), the coefficient plays a crucial role. The coefficient is the numerical part that is multiplied by the variable or variables. For this monomial, the coefficient is 5.

Here are some simple points to identify a coefficient in any monomial:
  • It is always a number.
  • The coefficient directly multiplies the variable part.
  • In a monomial like \(t^2\), without a visible coefficient, the coefficient is implicitly 1.
  • A negative sign, as in \(-5t^2\), is part of the coefficient, making it -5.
Understanding coefficients is essential because they help in simplifying expressions and solving equations that involve monomials.
Defining the Degree of a Monomial
The degree of a monomial is important for understanding its polynomial hierarchy. It tells us about the power of the variable.
  • The degree of a monomial is the sum of the exponents of all its variables.
  • In a simple monomial like \(5t^2\), the degree is 2, as there is only one variable \(t\) raised to the power of 2.
  • For a more complex monomial such as \(3x^2y^3\), the degree would be 5, calculated as \(2 + 3\).
  • A constant, like 7, is a monomial with a degree of 0 because there is no variable part.
Knowing the degree helps in adding or subtracting monomials, as only monomials of the same degree can be combined.
Exploring Exponents in Monomials
Exponents indicate the number of times a variable is multiplied by itself. In \(5t^2\), 2 is the exponent of variable \(t\).
  • An exponent is always written as a superscript next to the variable.
  • It shows repeated multiplication, so \(t^2\) means \(t \times t\).
  • A variable without an more explicit exponent, \(m\), implicitly has an exponent of 1, \(m^1\).
  • Exponents affect the calculation of the degree of a monomial, as we add them up for each variable.
Exponents are fundamental in determining both the shape and the operation involving monomials. Knowing how to manipulate exponents enables you to simplify expressions and carry out algebraic operations more effectively.

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Most popular questions from this chapter

State the conjugate of each binomial. $$-u+4$$

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