Problem 16 - Algebra Form and Function

Question

Give the leading term. $$ 12-3 x^{5}-15 x^{3} $$

Step-by-Step Solution

Answer

Answer: \(-3x^5\)

Step 1: Identify the terms in the given polynomial

Each term in the polynomial is separated by a plus (+) or minus (-) sign. In the given polynomial, we have the following terms: 1. 12 2. -3x^5 3. -15x^3

Step 2: Determine the degree of each term

The degree of a term is the power of the variable x in that term. Let's find the degree of each term: 1. 12 has a degree of 0 (constant term) 2. -3x^5 has a degree of 5 3. -15x^3 has a degree of 3

Step 3: Identify the leading term

The leading term is the term with the highest degree. In our case, the term with the highest degree is -3x^5 with a degree of 5. The leading term of the given polynomial is \(-3x^5\).

Key Concepts

Polynomial

A polynomial is a mathematical expression consisting of variables, coefficients, and exponents. These expressions are composed of terms separated by addition or subtraction. Each term in a polynomial may have:

• A coefficient, which is a numerical factor.
• A variable, commonly denoted as \(x\).
• An exponent, indicating the power to which the variable is raised.

For example, in the polynomial \(12 - 3x^5 - 15x^3\), each component separated by a plus (+) or minus (-) is a term. Understanding these components is crucial, as they establish the framework for further exploring terms' relationships and functions within the polynomial.

Degree of a Term

The degree of a term in a polynomial is determined by the exponent of its variable. This indicates the power to which each variable is raised. Here are key points to remember:

• A constant term, such as \(12\), with no variable, automatically has a degree of 0.
• A term like \(-3x^5\) has a degree of 5 because the variable \(x\) is raised to the fifth power.
• Another term, such as \(-15x^3\), has a degree of 3, corresponding to the cube of \(x\).

The degree helps in identifying relationships among terms and is essential for finding the leading term, as it directly corresponds to the term's priority within the polynomial.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. Spotting this correctly requires several steps:

• First, identify the term with the highest degree, known as the leading term.
• In the polynomial \(12 - 3x^5 - 15x^3\), the term \(-3x^5\) has the highest degree, which is 5.
• The coefficient of \(-3x^5\) is \(-3\), making \(-3\) the leading coefficient for this polynomial.

Understanding the leading coefficient is vital because it affects the behavior of the polynomial function graphically, especially as \(x\) becomes larger or smaller.