Problem 18 - Precalculus : Building Concepts and Connections

Question

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$g(t)=\frac{1}{t}$$

Step-by-Step Solution

Answer

The given function \(g(t) = \frac{1}{t}\) is not a polynomial function because it has a negative exponent.

Step 1: Understand Polynomial Function

A polynomial function has the general form \(f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where the \(a_i\) are coefficients and n is a non-negative integer. The highest power/exponent of x is called the degree of the polynomial. Polynomials cannot contain variables in the denominator, that is, negative exponents.

Step 2: Identify the Given Function

We're given the function \(g(t) = \frac{1}{t}\). This function may also be written as \(g(t) = t^{-1}\).

Step 3: Compare the Function with Polynomial Function

It is evident that the function contains a variable in the denominator that results in a negative exponent when rewritten. Negative exponents are not allowed in polynomial functions.

Step 4: Reason the Function is Not a Polynomial

Because polynomial functions cannot contain negative exponents and the function \(g(t) = \frac{1}{t} = t^{-1}\) does, it is not a polynomial function.

Key Concepts

Polynomial Degree

When studying polynomial functions, one important aspect to understand is the concept of the degree of a polynomial. The degree of a polynomial is defined as the highest power of the variable within the function. Each term in a polynomial can be expressed in the form \(a_ix^i\), where \(a_i\) are coefficients and \(i\) is a non-negative integer exponent that signifies the power of \(x\).

• The degree gives us a quick idea of the function’s shape and growth rate.
• A polynomial with a degree of 3, for example, is called a cubic polynomial, whereas a degree 2 polynomial is called a quadratic polynomial.

When evaluating a polynomial, identifying the term with the highest degree is crucial as it dictates the polynomial’s classification. This information assists in understanding how the function behaves as its variable approaches positive or negative infinity.

Negative Exponents

Negative exponents in mathematical expressions often lead to confusion, particularly in the context of polynomial functions. A polynomial function should not have any terms with negative exponents, because these represent division by the variable, which is not permissible in polynomials.

• Negative exponents mean that the base of the expression is in the denominator. For example, \(x^{-1}\) is equivalent to \(\frac{1}{x}\).
• Because a polynomial cannot include division by variables, expressions like \(t^{-1}\) cannot be part of a polynomial function.

Understanding why negative exponents are disallowed in polynomial functions is essential, as it helps in distinguishing non-polynomial expressions. Recognizing these restrictions simplifies identifying whether a given function, such as \(g(t) = \frac{1}{t}\), is a polynomial or not.

Non-negative Integers

Non-negative integers are a fundamental concept in mathematics, integral to the understanding of polynomial functions. Non-negative integers include all whole numbers starting from zero (0, 1, 2, 3,...), and they are prominently used as exponents in polynomial terms.

• In a polynomial function, the exponents of the variable must be non-negative integers. This requirement ensures that the polynomial only consists of terms like \(x^0, x^1, x^2, etc.\)
• This property of having non-negative integer exponents distinguishes polynomials from other types of functions and maintains their algebraic integrity.

When working through polynomial functions, ensuring each term has a non-negative integer exponent is key to correctly identifying them as true polynomials. This understanding clarifies why an expression involving \(t^{-1}\) is not considered a polynomial.