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Simplifying algebraic expressions often involves combining like terms, an essential step in solving equations and understanding algebra. Like terms are terms within an expression that have the same variable raised to the same power. For instance, in the expression \(2x^2 + 3x - 5 + 4x^2 - 2x\), the like terms \(2x^2\) and \(4x^2\) can be combined, as can the terms \(3x\) and \( -2x\).

When combining like terms, we simply add or subtract the coefficients (the numerical part of the terms) and keep the variable part unchanged. This process condenses the expression, making it easier to work with. In our example, the simplified expression would be \(6x^2 + x - 5\). By practicing the skill of combining like terms, students gain agility in managing more complex algebraic operations.

The distributive property, also known as the distributive law of multiplication over addition or subtraction, is a cornerstone of algebra. It states that for any numbers \(a\), \(b\), and \(c\), the equation \(a * (b + c) = a * b + a * c\) holds true. This property also applies when we distribute a negative sign, as in the exercise.

When faced with an expression like \( -2.6x^{2} + 49x + 3994 - ( -0.6x^{2} + 7x + 2412 )\), we apply the distributive property to remove the parentheses by distributing the negative sign, effectively changing the signs of the terms inside the parentheses. Thus, the expression becomes \( -2.6x^{2} + 49x + 3994 + 0.6x^{2} - 7x - 2412 \), setting up the expression for further simplification. Mastering the distributive property allows students to manipulate and simplify algebraic expressions with ease.

A polynomial is an algebraic expression consisting of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Polynomials are classified by their degree, which is the highest power of the variable in the expression. Examples include linear polynomials (degree 1), quadratic polynomials (degree 2), and cubic polynomials (degree 3).

The exercise provided involves a quadratic polynomial, as the highest degree of the variable \(x\) is 2. The process of simplifying polynomials involves combining like terms and using the distributive property, as previously discussed. The result is a simplified polynomial that is easier to evaluate or use in further calculations. Being familiar with polynomials and their properties is critical for success in algebra and higher-level mathematics.