Problem 14 - College Algebra with Modeling and Visualization

Question

Identify the degree and leading coefficient of the polynomial. $$-9 y^{4}+y^{2}+5$$

Step-by-Step Solution

Answer

Degree is 4, and leading coefficient is -9.

Step 1: Identify Terms

First, we need to list out all the terms in the polynomial: \(-9y^4\), \(y^2\), and \(5\).

Step 2: Determine the Degree

The degree of a polynomial is the highest power of the variable in the polynomial. Here, the degrees of the terms are:- \(-9y^4\): degree 4- \(y^2\): degree 2- \(5\): degree 0The highest degree among these is 4, so the degree of the polynomial is 4.

Step 3: Find the Leading Term

The leading term is the term with the highest degree. In this polynomial, \(-9y^4\) is the term with the highest degree, which is 4.

Step 4: Determine the Leading Coefficient

The leading coefficient is the coefficient of the leading term. For the term \(-9y^4\), the coefficient is \(-9\). Thus, the leading coefficient is \(-9\).

Key Concepts

Leading Coefficient

Understanding the leading coefficient of a polynomial is crucial for analyzing the polynomial's behavior. In any polynomial, the leading term is the one with the highest degree (the largest exponent of the variable). The leading coefficient is simply the numerical factor associated with this leading term.

Let's consider the polynomial \(-9y^4 + y^2 + 5\). Here, the leading term is \(-9y^4\). Hence, the leading coefficient is \(-9\).

The leading coefficient is significant because it affects the shape and direction of the graph of the polynomial. For instance:

• A positive leading coefficient typically implies that as \(x\) approaches infinity, so does \(f(x)\).
• A negative leading coefficient usually suggests the polynomial decreases to negative infinity as the variable grows.

Grasping the leading coefficient helps predict the end behavior of the polynomial's graph.

Polynomial Terms

Polynomials are composed of terms, and each term consists of a coefficient, a variable, and an exponent. In the polynomial \(-9y^4 + y^2 + 5\), there are specific parts to recognize:

• \(-9y^4\) is a term with a coefficient of \(-9\), variable \(y\), and exponent \(4\).
• \(+y^2\) is another term where the coefficient is \(1\), the variable is \(y\), and the exponent is \(2\).
• \(+5\) is a constant term with no variable, which can be rewritten as \(5y^0\) to denote it has a degree of zero.

Each term represents a part of the total polynomial's value. A polynomial's terms combine according to the rules of algebra to form expressions that are used widely in mathematical computations. Recognizing each term helps in organizing a polynomial and setting the groundwork for further calculations.

Degree of Polynomial

The degree of a polynomial is a key concept that denotes the greatest power of the variable present in the polynomial. It influences the polynomial's shape and is useful for approximating its general behavior over time.

To determine the degree, examine all the exponents in the polynomial. For the polynomial in question, \(-9y^4 + y^2 + 5\), you check:

• \(-9y^4\) has a degree of 4.
• \(+y^2\) has a degree of 2.
• \(+5\) has a degree of 0, as it is a constant and does not have a variable component.

The highest degree in these terms is 4. Therefore, the degree of the polynomial is 4.

Understanding the degree of a polynomial aids in predictions about the polynomial's graph, such as the potential number of roots and the general type of curve it represents when plotted.