Question

Decide whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. \(p(x)=2 x-5 x^3+9 x^2+\sqrt[4]{x}+1\)

Short answer

The function \(p(x)=2 x-5 x^3+9 x^2+\sqrt[4]{x}+1\) is not a polynomial function because it contains an expression with a fractional exponent.

Step-by-step solution

Identification of a Polynomial Function

To identify if the given function is a polynomial we must look at the exponents of the variable \(x\). All exponents in a polynomial function must be nonnegative integers(representing whole numbers). In this case, we see that all the variable \(x\) in the given function - \(p(x)=2 x-5 x^3+9 x^2+\sqrt[4]{x}+1\) have exponents of whole numbers except for the term \(\sqrt[4]{x}\) which amounts to \(x^{1/4}\), which has a fractional exponent, and does not meet the requirement of a polynomial function.

Decision

Because the function contains an expression with a fractional exponent, \(x^{1/4}\), the function \(p(x)=2 x-5 x^3+9 x^2+\sqrt[4]{x}+1\) is not a polynomial function. So, it is not required to write this in standard form or to state its degree, type, or leading coefficient.

Key concepts

Polynomial Degree

The degree of a polynomial is a key characteristic. It indicates the highest power of the variable in the polynomial. To determine the degree, pick the term with the largest exponent. For example, in the polynomial \(5x^3 + 3x^2 - 2x + 7\), the term \(5x^3\) has the highest exponent, which is 3. Hence, the degree of this polynomial is 3.

This matters because the degree can affect the shape of the graph of the polynomial and tells us the maximum number of real roots or solutions it can have. In general, a polynomial of degree \(n\) can have at most \(n\) x-intercepts.

• Degree 1: Linear
• Degree 2: Quadratic
• Degree 3: Cubic
• Degree 4: Quartic

Exponents

Exponents are an essential part of polynomial functions. They show how many times a number, known as the base, is multiplied by itself. In a polynomial, the exponents must be nonnegative integers. This means numbers like 0, 1, 2, and so forth.

Nonnegative integer exponents ensure that a polynomial graph is smooth and continuous. If you have an exponent like \(1/4\) (as in \(\sqrt[4]{x}\)), then it doesn't qualify as a polynomial term. This is because fractional exponents introduce breaks and undefined zones for negative values in the graph, making it not a typical polynomial curve. Therefore, all terms in a polynomial should have whole number exponents.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial, often clarifying the polynomial's behavior as the variable values increase or decrease without bound. In a polynomial written in standard form, the leading term is the first term.

For example, consider the polynomial \(-5x^3 + 4x^2 + 7\). Here, the leading term is \(-5x^3\), and the leading coefficient is \(-5\). This leading coefficient can also indicate the end behavior of the polynomial:

• If the leading coefficient is positive, the polynomial's ends shoot upwards.
• If it's negative, the ends fall.

This concept is crucial for predicting the long-term, or asymptotic, behavior of the polynomial function's graph.

Standard Form

Writing a polynomial in standard form means arranging the terms in order of descending exponents. This arrangement allows for easy identification of the leading term and its coefficient, and it simplifies analyzing the function.

For example, for the polynomial \(2x + 5x^3 - 3x^2\), the standard form is \(5x^3 - 3x^2 + 2x\). This rearrangement is based on organizing the terms from the highest exponent to the lowest.

• Highest exponent term first
• Middle exponent terms follow
• Constant terms last (exponent 0)

While not every function can be converted into a polynomial's standard form, when it can, this format allows us to quickly understand its degree, leading coefficient, and overall structure.