Problem 15 - Precalculus : Building Concepts and Connections

Question

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$f(x)=-x^{3}+3 x^{3}+1$$

Step-by-Step Solution

Answer

Yes, the function \(f(x)=-x^{3}+3 x^{3}+1\) is a polynomial function and its degree is 3

Step 1: Definition of a Polynomial Function

A polynomial function is a function such as the quadratic function, the cubic function, the quartic function and so on, in which the variable's highest power determines the name of the function. A polynomial function is composed of terms, where a term is the product of a number called a coefficient and some positive, integral powers of the variable.

Step 2: Forming a Polynomial Function

Looking at our function \(f(x)=-x^{3}+3 x^{3}+1\), this is composed of terms. Adding the like terms -1x^3 and 3x^3, we have : \(f(x)=2x^{3}+1\) . This is a clear example of a polynomial function as it meets the definition provided above.

Step 3: Finding the Degree of the Polynomial Function

The degree of the polynomial function is the highest power of the variable in the function. For our function, \(f(x)=2x^{3}+1\), the highest power of the variable, x, is 3. Hence, the degree of the polynomial function is 3.

Key Concepts

Degree of a Polynomial

When we talk about the degree of a polynomial, we are referring to the highest power of the variable in any given term of the polynomial. It's important to realize that a polynomial can have several terms, but the term with the highest exponent is what determines its degree. For example, in the polynomial function \(f(x) = 4x^5 + 2x^3 - x + 7\), the degree is 5 because the term \(4x^5\) has the highest power of \(x\), which is 5. To identify the degree of a polynomial, one must simply look for the term with the largest exponent.

In the exercise provided, let's recall that the initial function was \(f(x)=-x^{3}+3x^{3}+1\), which simplified to \(f(x)=2x^{3}+1\) by combining like terms. The term \(2x^{3}\) has the highest exponent, indicating that our polynomial is indeed a cubic function and its degree is 3.

Coefficient

A coefficient in a polynomial function is a numerical factor that multiplies the variable(s). Coefficients can be positive, negative, or zero, and they can be integers, fractions, or even irrational numbers. For instance, in the polynomial \(2x^{2} + 3x - 1\), the coefficients are 2, 3, and -1 for the respective terms \(x^{2}\), \(x\), and the constant term -1.

In the context of the exercise solution, we initially encountered the coefficients -1 and 3 in the terms \( -x^{3} \) and \(3x^{3} \) respectively. However, after simplification, the resulting polynomial is \(f(x)=2x^{3}+1\), and here, the number 2 is the coefficient of the term \(x^{3}\), while the number 1 is the coefficient of the constant term in the polynomial.

Cubic Function

A cubic function is a particular type of polynomial function where the highest degree is 3. This means the function will include a term with the variable raised to the third power. It has the general form \(ax^{3} + bx^{2} + cx + d\), where \(a\), \(b\), \(c\), and \(d\) are coefficients, and \(a \eq 0\). A key characteristic of cubic functions is their potential to have two turning points and thus display a characteristic 'S' shaped curve when graphed.

In the exercise provided, after combining like terms, we reached the function \(f(x)=2x^{3}+1\). This is indeed a cubic function, simplified yet following the general form. It has a single term with the variable to the third power, a coefficient of 2, and a constant term. This means if graphed, we could expect an 'S' shaped curve representative of cubic functions.