Problem 21 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The function \(f(x) = -(x+1)^{3}\) is a polynomial function. The degree of the polynomial is 3.

Step 1: Identify If The Given Function Is A Polynomial

To start, identify whether the given function is a polynomial. In this case, the function is \(f(x) = -(x+1)^{3}\). This function only has variables that are raised to whole number exponents, and thus falls into the category of polynomial functions.

Step 2: Identify The Degree Of The Polynomial

The degree of the polynomial is determined by the highest exponent when the polynomial is in its simplest form. In this polynomial, the highest exponent is 3, which means that the degree of the polynomial is 3.

Key Concepts

Degree of a Polynomial

Understanding the degree of a polynomial is fundamental in the study of polynomial functions. The degree is essentially the highest power of the variable in the polynomial when it is expressed in its standard form. In the given exercise, \(f(x) = -(x+1)^{3}\), to find the degree, we expand the expression to its simplest form, which would look like \( ax^n + bx^{n-1} + ... + k \), where \( a, b, ..., k \), are constants, and \( n \) is a whole number.

For our example, after expanding \( (x+1)^3 \), we have a cubic term as the highest power of x, thus, the degree of the polynomial is 3. This is a critical concept because the degree can tell us the number of roots and the basic shape of the graph of the polynomial. For instance, a third-degree polynomial will have, at most, three real roots and will have two turns on its graph.

Identifying Polynomial Functions

To identify polynomial functions, there are specific characteristics we look for. A polynomial function will have terms that are made up of constants, variables, or the product of constants and variables. Each term is also known as a monomial and the variables in a polynomial are raised to positive integer powers or are to the zeroth power (which equals one).

In the exercise \(f(x) = -(x+1)^{3}\), we see that the function is made up of a variable \(x\) that is raised to a whole number exponent within the expression \( (x+1)^3 \). If a function involves variables raised to negative or fractional exponents, or variables in the denominator, it is not classified as a polynomial function. Ensuring that all powers are non-negative integers is essential in verifying that a function is polynomial.

Exponents

Exponents play a central role in classifying and working with polynomial functions. An exponent tells us how many times to use the base as a factor in a product. For example, in \(x^3\), the base is \(x\) and the exponent is 3, indicating that \(x\) is used as a factor three times: \(x * x * x\).

In the context of polynomial functions, we are mainly concerned with whole number exponents. They dictate the degree of the polynomial and influence the curvature and number of turning points on the graph of the function. Hence, understanding how to work with exponents is key to mastering polynomial functions. When simplifying expressions like \( (x+1)^3 \), we use the binomial theorem or the distributive property to express it as a polynomial where the exponents indicate the degrees of individual terms.