Problem 12 - College Algebra

Question

Find the degree and leading coefficient for the given polynomial. $$-3 x$$

Step-by-Step Solution

Answer

Degree: 1, Leading Coefficient: -3.

Step 1: Identify Polynomial Terms

The given expression is \(-3x\).This is a polynomial with a single term, also known as a monomial.

Step 2: Determine the Degree of the Polynomial

The degree of a polynomial is determined by the highest power of the variable. In \(-3x\), the variable \(x\) is raised to the power of 1. Therefore, the degree of the polynomial is 1.

Step 3: Identify the Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree.In \(-3x\), the term with the highest degree is \(-3x\) itself, and its coefficient is \(-3\). Thus, the leading coefficient is \(-3\).

Key Concepts

Degree of a Polynomial

Polynomials are mathematical expressions that consist of variables and coefficients. The degree of a polynomial is a crucial concept that helps in understanding the behavior or nature of the polynomial. It is primarily determined by the highest power of the variable in any of the terms of the polynomial. For instance, in a single term polynomial like
\(-3x\), the variable \(x\) is raised to the power of 1. Hence, the degree of the polynomial here is 1.

• For example, in a polynomial \(5x^3 - 4x^2 + 2\), the degree is 3 because the highest power of the variable \(x\) is 3.
• In a polynomial with more than one variable, such as \(-3xy^2 + 2x^2y - 6\), the degree would be determined by the highest sum of the exponents in any term, so the degree is 3. (From the term \(x^2y\)).

Understanding the degree is essential as it influences how the polynomial behaves when graphed and how it reacts to different mathematical operations.

Leading Coefficient

In polynomials, coefficients play a critical role as they multiply the power of the variable. The leading coefficient is the coefficient of the term with the highest degree. This is important because it influences the shape and direction of the polynomial graph. In a single monomial like \(-3x\), the highest degree term is \(-3x\) itself, and the coefficient of this term is \(-3\).

• For example, in the polynomial \(4x^5 - 2x^4 + 7\), the leading coefficient is 4 since the term \(4x^5\) has the highest degree of 5.
• If you encounter a polynomial like \(x^3 - 5x^2 + x\), the implicit coefficients are vital: the leading coefficient here is 1 for \(x^3\).

Remember, the leading coefficient is helpful for determining polynomial end behavior, providing insights into how the graph of the polynomial functions as \(x\) approaches either positive or negative infinity.

Monomials

A monomial is the simplest type of polynomial, consisting of just one term. It is an algebraic expression that includes numbers, variables, or both that are multiplied together to form a single term. The term \(-3x\) is a monomial.

• Monomials can include numbers, variables, or both, such as \(7y\) or \(-4z^2\).
• They do not have plus or minus signs separating different terms since there is just one term.

Monomials are foundational elements that make up more complex polynomials. Understanding and identifying monomials can simplify more advanced concepts in algebra and calculus, as each term in a polynomial behaves like an individual monomial.