To determine why [tex]\( x^2 \)[/tex] is a polynomial but [tex]\( \frac{1}{x^2} \)[/tex] is not, let's first define what a polynomial is.
A polynomial is an expression consisting of variables (also called indeterminates), coefficients, and exponents, combined using only addition, subtraction, and multiplication. The key characteristics of polynomials are:
1. The exponents of the variables must be non-negative integers (i.e., 0, 1, 2, 3, ...).
2. The coefficients of the terms can be any real or complex number.
### Example of a Polynomial
Consider the expression [tex]\( x^2 \)[/tex].
1. The variable in this expression is [tex]\( x \)[/tex].
2. The exponent of [tex]\( x \)[/tex] is [tex]\( 2 \)[/tex], which is a non-negative integer.
3. There are no negative exponents or fractional exponents involved.
Thus, [tex]\( x^2 \)[/tex] satisfies the criteria of a polynomial.
### Example of a Non-Polynomial
Now consider the expression [tex]\( \frac{1}{x^2} \)[/tex].
1. This expression can be rewritten as [tex]\( x^{-2} \)[/tex] to express it with a negative exponent.
2. The exponent of [tex]\( x \)[/tex] in this form is [tex]\( -2 \)[/tex], which is a negative integer.
3. Polynomials are not allowed to have negative or fractional exponents, only non-negative integers.
Since [tex]\( \frac{1}{x^2} \)[/tex] or [tex]\( x^{-2} \)[/tex] involves a negative exponent, it does not meet the criteria for being a polynomial.
### Summary
In summary:
- [tex]\( x^2 \)[/tex] is a polynomial because it has an exponent that is a non-negative integer.
- [tex]\( \frac{1}{x^2} \)[/tex] or [tex]\( x^{-2} \)[/tex] is not a polynomial because it has a negative exponent, which is not allowed in polynomials.
Hence, [tex]\( x^2 \)[/tex] is classified as a polynomial, whereas [tex]\( \frac{1}{x^2} \)[/tex] is not.
