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Polynomials are expressions consisting of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Unlike regular numbers, polynomials can be much more complex due to the presence of variables.
An example of a simple polynomial is \( 3x^2 + 2x - 5 \). Here, \( 3x^2 \), \( 2x \), and \( -5 \) are terms of the polynomial. The highest exponent of the variable determines the degree of the polynomial. In this case, the degree is 2 since the highest exponent is 2.

In a fraction, the numerator and denominator are two critical parts. The numerator is the top part of the fraction, while the denominator is the bottom part. In rational expressions, both the numerator and denominator are polynomials.
For instance, in the rational expression \[\frac{x^2 - 4}{x + 2} \], \( x^2 - 4 \) is the numerator, and \( x + 2 \) is the denominator. It's crucial to note that the denominator cannot be zero because division by zero is undefined.

Simplifying expressions involves reducing them to their simplest form. For rational expressions, this means factoring both the numerator and the denominator to cancel out any common factors, if possible.
Take the expression \[ \frac{x^2 - 4}{x + 2} \] as an example. We can simplify it by recognizing that \( x^2 - 4 \) is a difference of squares, which can be factored as \( (x - 2)(x + 2) \). This gives us the new expression \[ \frac{(x - 2)(x + 2)}{x + 2} \]. Here, \( (x + 2) \) in the numerator and denominator cancel each other, simplifying the rational expression to \( x - 2 \).
This process of factoring and canceling is key for simplifying rational expressions. Just remember to ensure the denominator is not zero after simplifying the expression.