Which of the following expressions is equivalent to 3\(x\left(x^{2} y+2 x y^{2}\right) ?\) F. \(3 x^{2} y+6 x y^{2}\) G. \(3 x^{3} y+6 x y^{2}\) H. \(3 x^{3} y+6 x^{2} y^{2}\) J. 5\(x^{4} y^{3}\) K. 9\(x^{4} y^{3}\)

The distributive property is a fundamental tool in algebra that allows you to multiply a single term by each term inside a set of parentheses. For example, given the expression \(a(b + c)\), the distributive property tells us to multiply \(a\) by \(b\) and \(a\) by \(c\), and then add the results together to get \(ab + ac\).

This property is very useful for simplifying more complex algebraic expressions, as in the ACT Math Practice problem we're considering. To apply it, you can 'distribute' the term outside the parenthesis to each term inside. In the exercise, \(3x\) is distributed to both \(x^2y\) and \(2xy^2\), resulting in a simplified expression. The distributive property makes handling polynomials and combining like terms manageable, making it a cornerstone for working through many high school and college-level math problems.

Simplifying expressions is the process of combining like terms and performing operations to condense an algebraic expression into its simplest form. This often involves applying the distributive property, as well as the associative and commutative properties of addition and multiplication. When simplifying, you look for terms that have the same variables raised to the same powers and then combine them.

For example, if you are given the expression \(2x + 3x^2 + x\), you would combine the like terms \(2x\) and \(x\) to get \(3x\) and then rewrite the expression as \(3x + 3x^2\). Simplifying expressions is not just a mechanical process; it requires an understanding of underlying mathematical principles to ensure that the expression remains equivalent throughout the simplification process.

Polynomial operations include addition, subtraction, multiplication, and division of polynomial expressions. These operations are built upon the distributive property, along with the commutative and associative laws. In the context of the ACT Math Practice problem, we are concerned with the multiplication of polynomials. To multiply polynomials, each term of the first polynomial must be multiplied by each term of the second polynomial.

As seen in the step-by-step solution, \(3x(x^2y + 2xy^2)\) involves multiplying a monomial by a binomial. Here, \(3x\) multiplies each term in the binomial. This requires an understanding of how to handle exponents during multiplication — that is, when you multiply terms with the same base, you add their exponents (e.g., \(x^m \cdot x^n = x^{m+n}\)). This process results in a new polynomial whose terms are then combined if possible, producing a simplified answer.