Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Find the degree and leading coefficient of the polynomial.\(-3 x^{4}+2 x^{2}-5\)

Step-by-Step Solution

Answer

The degree of the polynomial is 4 and the leading coefficient is -3.

Step 1: Find the highest degree

Look at the polynomial \(-3 x^{4}+2 x^{2}-5\) and identify the term with the highest power of the variable x. This can be identified by identifying the largest exponent, which in this case is the term \(-3 x^{4}\) with an exponent of 4. This means that the degree of the polynomial is 4.

Step 2: Find the leading coefficient

The leading coefficient is the coefficient of the term with the highest degree. In this case, the coefficient of term \(-3 x^{4}\) is -3. Therefore, the leading coefficient of the polynomial is -3.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is a key concept that helps us understand many characteristics of polynomial functions. In any polynomial, the degree is determined by the term with the highest power of the variable. For example, in the polynomial \[-3x^4 + 2x^2 - 5\], the term \(-3x^4\) has the highest exponent, which is 4. Thus, the degree of this polynomial is 4.

Knowing the degree of a polynomial gives us insights into the function's behavior. It tells us about the maximum number of roots (real roots and complex roots combined) the function can have, and also the overall shape of its graph.

Key points to remember about polynomial degree:

• The degree is the largest exponent of the variable in the polynomial.
• Only terms with non-zero coefficients are considered when identifying the degree.
• Constantsi, like \(-5\) in this example, can also be viewed as \(x^0\), but they do not impact the degree if higher powers are present.

Leading Coefficient

The leading coefficient is another vital aspect to understand when dealing with polynomials. It is the coefficient of the term that has the highest degree in the polynomial.

In our example, the polynomial is \[-3x^4 + 2x^2 - 5\]. The term \(-3x^4\) has the highest degree, which makes \(-3\) the leading coefficient. This coefficient is crucial in determining several attributes of the polynomial, such as the end behavior of its graph.

Important details about the leading coefficient:

• It is the numerical part of the term with the highest degree.
• The sign of the leading coefficient (positive or negative) affects the direction of the graph's arms as they approach infinity.
• The magnitude of the leading coefficient can affect how "wide" or "narrow" the graph appears.

Exponents in Polynomials

Exponents are fundamental components of polynomials that influence both the degree and the shape of the polynomial's curve. In any polynomial expression, each term is expressed in the form \(ax^n\), where \(a\) is the coefficient and \(n\) is the exponent.

For the polynomial \(-3x^4 + 2x^2 - 5\), note that:

• The term \(-3x^4\) has an exponent of 4.
• \(2x^2\) involves \(x\) raised to the power of 2.
• The constant \(-5\) can be thought of as \(-5x^0\).

Exponents affect several characteristics of polynomials:

• The value of the exponent indicates the term's degree and contributes to the overall degree of the polynomial.
• Even and odd exponents have different effects on the "shape" of polynomial graphs.