Is ( x^{frac{5}{2}} - 3x^{frac{3}{2}} - 2x^{frac{1}{2}} ) a Polynomial? Understanding the Definition and Degree
In the context of algebra, polynomials play a vital role in mathematical analysis. Understanding what constitutes a polynomial and how to determine its degree is a fundamental aspect of algebraic theory. In this article, we will explore the expression ( x^{frac{5}{2}} - 3x^{frac{3}{2}} - 2x^{frac{1}{2}} ), examining whether it qualifies as a polynomial and, if so, determining its degree.
Defining Polynomials
A polynomial in a single variable ( x ) is typically expressed as a sum of terms, each of which is a product of a constant coefficient and a variable raised to a non-negative integer power (the exponent).
Mathematically, a polynomial can be written as:
[ a_n x^n a_{n-1} x^{n-1} cdots a_1 x a_0 ]
where ( a_i ) (for ( i 0, 1, ldots, n )) are constants and ( n ) is a non-negative integer.
Analysis of the Given Expression
Let's analyze the expression ( x^{frac{5}{2}} - 3x^{frac{3}{2}} - 2x^{frac{1}{2}} ). To determine if this is a polynomial, we must check if all exponents of ( x ) are non-negative integers.
The exponents in the expression are:
( frac{5}{2} ) ( frac{3}{2} ) ( frac{1}{2} )None of these exponents are non-negative integers. This immediately tells us that the expression is not a polynomial in the variable ( x ).
Transforming the Expression
Although the expression is not a polynomial in ( x ), it can be transformed into a polynomial form by introducing a new variable. Consider the variable ( u x^{frac{1}{2}} ). Then, we can rewrite the expression in terms of ( u ):
[ u^5 - 3u^3 - 2u ]
This new expression is clearly a polynomial in terms of ( u ).
Conclusion and Further Insights
Returning to our original expression ( x^{frac{5}{2}} - 3x^{frac{3}{2}} - 2x^{frac{1}{2}} ), we conclude that it is not a polynomial in ( x ) since the exponents are not non-negative integers.
For further study, you can refer to the Wikipedia article Degree of a Polynomial for more advanced discussions and examples of polynomial functions.
Keywords: polynomial, degree, positive integer exponents