Problem 14 - College Algebra

Question

Find the degree and leading coefficient for the given polynomial. $$-2 x^{2}-3 x^{5}+x-6$$

Step-by-Step Solution

Answer

The degree is 5, and the leading coefficient is -3.

Step 1: Identify the Terms

Write down each term of the polynomial separately: 1. -2x^2 2. -3x^5 3. x 4. -6.

Step 2: Find the Degree of Each Term

Determine the degree of each term by looking at the exponent of the variable. - 2x^2 has degree 2. - -3x^5 has degree 5. - x has degree 1. - The constant -6 has degree 0.

Step 3: Identify the Degree of the Polynomial

The degree of the polynomial is the highest degree among its terms. In this polynomial, the term -3x^5 has the highest degree which is 5. Therefore, the degree of the polynomial is 5.

Step 4: Determine the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree. For -3x^5, the coefficient is -3. Thus, the leading coefficient is -3.

Key Concepts

Leading Coefficient

The leading coefficient is a crucial part of a polynomial expression. It refers specifically to the coefficient of the term with the highest degree. In our problem, the term with the highest degree is \(-3x^5\). The degree of this term is 5, which is higher than any other term in this polynomial. Therefore, the leading coefficient for this polynomial is \(-3\). Remember, the leading coefficient determines the polynomial's end behavior and helps in graphing its tendencies. It can be positive or negative, affecting the direction the ends of the graph point.

Polynomial Terms

Polynomials are composed of terms. Each term can be a variable, a constant, or a product of both. In the polynomial \(-2x^{2} - 3x^{5} + x - 6\), there are four distinct terms:

• \(-2x^2\)
• \(-3x^5\)
• \(x\)
• \(-6\)

Each term can have coefficients, variables with exponents, and constants. Familiarity with these terms will help you understand how they interact within the polynomial. You identify each term by looking for pieces of the expression divided by a plus or minus sign.

Exponents

Exponents are the powers to which the variable is raised in a term. They determine the degree of each term in a polynomial. In the polynomial provided:

• \(-2x^2\) has an exponent of 2.
• \(-3x^5\) has an exponent of 5.
• \(x\), which is the same as \(x^1\), has an exponent of 1.
• The constant \(-6\) has an implicit exponent of 0, as it does not multiply any variable.

The term with the highest exponent defines the polynomial's degree. In this case, the polynomial's degree is 5 because the term \(-3x^5\) has the highest exponent.

Constant Terms

Constant terms are the terms in a polynomial that do not contain variables, only a consistent value. In our example, the term \(-6\) is the constant term. Since it doesn’t have a variable attached, its degree is always 0. Constant terms are crucial because they can affect the vertical shift of a polynomial's graph on a coordinate plane. They're often the simpler part of a polynomial but play an essential role in the broader expression. A constant is sometimes seen as the 'starting point' of the polynomial when evaluating it for any variable's value.