Question

USING STRUCTURE Determine whether the function is a polynomial function. If so, write it in standard form and state its degree, type, and leading coefficient. $$ f(x)=5 x^3 x+\frac{5}{2} x^3-9 x^4+\sqrt{2} x^2+4 x-1-x^{-5} x^5-4 $$

Short answer

The function \( f(x)=5x^3x + {\frac{5}{2}}x^3 - 9x^4 + \sqrt{2}x^2 + 4x - 1 -x^{-5}x^5 - 4 \) is not a polynomial function because it contains the term \(x^{-5}x^5\), which includes a negative exponent. Polynomial functions must only have positive, whole number exponents.

Step-by-step solution

Identify whether the function is a polynomial

Look at the given function \( f(x)=5x^3x + {\frac{5}{2}}x^3 - 9x^4 + \sqrt{2}x^2 + 4x - 1 -x^{-5}x^5 - 4 \). The term \(x^{-5}x^5\) in this function is the reason why this function can't be considered a polynomial function because it involves a negative exponent. A polynomial functions must have positive, whole number exponents. This is the deciding factor of this exercise.

No further steps necessary

Since the function \( f(x) \) was identified not to be a polynomial in the first step, no other steps, such as determining degree, type and leading coefficient, are necessary.

Key concepts

Standard Form of Polynomial

Understanding the standard form of a polynomial is essential in algebra. A polynomial in standard form is written with its terms in descending order of exponent from left to right. The exponents are positive integers, and the coefficients can be any real numbers. For instance, a cubic polynomial written in standard form would be something like \( ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) represent the coefficients.

Looking at the original exercise, if we were to disregard the term with negative exponent for a moment, the standard form would require rewriting the function after combining like terms. So, considering \( f(x) = 5x^4 + \frac{5}{2}x^3 - 9x^4 + \sqrt{2}x^2 + 4x - 1 - 4 \), we'd combine \( 5x^4 \ and -9x^4 \), to get the polynomial in the form \( -4x^4 + \frac{5}{2}x^3 + \sqrt{2}x^2 + 4x - 5 \). However, the presence of the \( x^{-5}x^5 \), or any other term with a negative exponent, disqualifies the expression from being considered a polynomial.

Degree of Polynomial

The degree of a polynomial is the highest power of the variable that occurs in the polynomial when it is written in its standard form. It gives us a way to classify polynomials and also predicts many of the polynomial's properties, including the number of possible real roots and the shape of its graph. In the exercise, if it were a polynomial, the degree would be determined by the term with the highest exponent that has a non-zero coefficient after combining like terms and disregarding any terms with negative or fractional exponents.

In this case, the term with the highest exponent would be \( -4x^4 \) before consideration of the non-polynomial term \( x^{-5}x^5 \). This would mean the function has a degree of 4, making it a quartic polynomial—had it not been for the presence of the term with the negative exponent, which is not allowed in polynomial expressions.

Leading Coefficient

The leading coefficient of a polynomial is the coefficient of the first term when the polynomial is written in standard form, i.e., the term with the highest power as we move from left to right. It is an important factor since it impacts the end behavior of a polynomial's graph. If the leading coefficient is positive, the polynomial will eventually rise to positive infinity as \( x \) goes to positive infinity, assuming the degree is even.

In the context of the problem, assuming we could ignore the term with the negative exponent, the leading coefficient would be found in front of the \( x^4 \) term (which should be the highest degree term), so the leading coefficient would have been -4. However, since the expression cannot be considered a polynomial, it technically does not have a leading coefficient in the context of polynomial functions.