Problem 1 - College Algebra with Modeling and Visualization

Question

Determine if the junction is a polynomial function. If it is, state its degree and leading coefficient a. $$ f(x)=2 x^{3}-x+5 $$

Step-by-Step Solution

Answer

The function is a polynomial of degree 3, with a leading coefficient of 2.

Step 1: Identify the Form of the Function

To determine if a function is a polynomial, we begin by examining its form. Each term should be a constant multiplied by a whole number power of the variable. The given function is \( f(x) = 2x^3 - x + 5 \). We see three terms: \( 2x^3 \), \(-x\), and \(5\). Each term fits the polynomial form.

Step 2: Verify Whole Number Powers

Ensure each term's exponent is a whole number. In the function \( f(x) = 2x^3 - x + 5 \), the exponents of \(x\) are 3 and 1, both of which are whole numbers. The constant term \(5\) can be seen as \(5x^0\), which also fits the criteria.

Step 3: Determine the Polynomial's Degree

The degree of a polynomial is the highest power of the variable \(x\) in the function. In \( f(x) = 2x^3 - x + 5 \), the highest power is 3. Therefore, the degree of the polynomial is 3.

Step 4: Identify the Leading Coefficient

The leading coefficient is the coefficient of the term with the highest power in the polynomial. In \( f(x) = 2x^3 - x + 5 \), the highest power is 3 and the corresponding term is \(2x^3\). Therefore, the leading coefficient is 2.

Key Concepts

Degree of a Polynomial

One fundamental characteristic of polynomials is their degree. The degree of a polynomial is defined as the highest power of the variable, often denoted as \( x \), in the expression. In simple terms, it's the largest exponent found in the polynomial.

When examining a polynomial like \( f(x) = 2x^3 - x + 5 \), we look for the term with the highest exponent. Here, we observe the term \( 2x^3 \). The exponent is 3, which means the degree of this polynomial is 3.

Understanding the degree helps you grasp the polynomial's behavior for large values of \( x \), as higher degree terms grow much faster. The degree also indicates the highest number of roots or zeros the polynomial can have, an essential concept in algebra.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the term with the highest degree in the expression. This term essentially guides the "direction" of the polynomial's graph as \( x \) extends toward the positive or negative infinity.

Let's take our example polynomial \( f(x) = 2x^3 - x + 5 \). The term with the highest degree is \( 2x^3 \), thus the leading coefficient is 2. Imagine this as the "weight" that tips the scale of how the polynomial behaves on a graph.

In practical terms, the leading coefficient influences how steep or flat a polynomial curve will be. Positive coefficients (like 2 in our example) lead to upward trends in graphs, while negative ones lead to downward trends as you venture further out along the \( x \)-axis.

Exponents in Polynomials

Within polynomial expressions, exponents play a crucial role in defining how each term behaves. In polynomials, the exponents are always non-negative whole numbers.

• The exponent tells us how many times the variable \( x \) is multiplied by itself in that term.
• For example, in \( 2x^3 \), the exponent 3 informs us that \( x \) appears three times as a factor.
• A special case is the constant term, like \( 5 \) in our polynomial \( f(x) = 2x^3 - x + 5 \), which can be seen as \( 5x^0 \) because \( x^0 = 1 \).

Adhering to whole number and non-negative rules ensures the expression remains a polynomial. If an exponent is fractional or negative, it would no longer qualify as a polynomial term.