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Polynomial simplification involves rewriting a polynomial in its simplest form. To simplify a given polynomial, you need to perform operations such as addition, subtraction, multiplication, and sometimes division.
The goal is to combine like terms and reduce the expression to its simplest form, without altering its value. For instance, in the exercise \((a^3 - 3a) - (1 - a - 2a^2)\), the first step in simplification is to manage the parentheses and apply distributive properties available.

This step can include distributing negative signs or multiplying terms inside the parentheses. After distribution, combine all duplicate elements, known as 'like terms', to form a single simpler expression. The result still maintains the polynomial's original value but in an easier-to-handle format.

Combining like terms is one of the fundamental steps in simplifying polynomials. Like terms are terms that include the same variable raised to the same power. For example, in the expression \(2a^2 + 3a - 4 - a + 5a^2\), the like terms are \(2a^2\) and \(5a^2\) because they both have the variable \(a\) raised to the power of 2.

To combine like terms, simply add or subtract the coefficients while keeping the variable part unchanged. For instance, combining \(2a^2\) and \(5a^2\) will result in \(7a^2\).
In our exercise, after distributing the negative sign: \(a^3 - 3a - 1 + a + 2a^2\), we identify the like terms: \(a^3\), \(2a^2\), \(-3a\), and \(a\).
By combining them: \(a^3 + 2a^2 + (-3a + a) - 1\), we get the simplified expression \(a^3 + 2a^2 - 2a - 1\).

The distributive property is a fundamental principle used in algebra. It assists in dealing with algebraic expressions involving parentheses. The distributive property states that \((a(b + c) = ab + ac)\).
This property is helpful when simplifying polynomials and combining like terms.

For example, in our given problem \((a^3 - 3a) - (1 - a - 2a^2)\), the first step uses the distributive property to manage the negative sign: \(a^3 - 3a - 1 + a + 2a^2\).
Distributing the negative sign ensures that each term inside the parenthesis is correctly simplified to its proper term outside the parenthesis, facilitating easier further simplifications. This is a key step in transforming the initial expression into a simpler, more manageable form.