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Understanding the distributive property is crucial in algebra, especially when dealing with polynomials. In simple terms, it allows us to multiply a single term by each term within a parenthesis. Take, for instance, the expression \(a(b+c)\). Applying the distributive property, we multiply \(a\) by \(b\) and then by \(c\), resulting in \(ab + ac\).

This very property is used during the first step of the exercise where \(x_{1}+x_{2}\) is multiplied by \(x_{1}+x_{3}\), expanding the expression to \(x_{1}^2 + x_{1}x_{3} + x_{2}x_{1} + x_{2}x_{3}\). Pay attention to how each term outside the parenthesis is distributed and multiplied by every term inside, which is the heart of the distributive property.

When simplifying algebraic expressions, combining like terms is an essential step towards a cleaner and more concise expression. But what are like terms? They're terms that have the exact same variable parts, making them compatible for combination. For example, \(2x\) and \(3x\) are like terms, and thus, together they amount to \(5x\).

In the exercise, after applying the distributive property, we identify and combine like terms which, in this case, are \(x_{1}x_{3}\) and \(x_{2}x_{1}\) that simplifies to \(2x_{1}x_{3}\) since \(x_{1}x_{3}\) and \(x_{2}x_{1}\) have the same variables. This step is crucial in polynomial simplification as it considerably reduces the complexity of the expression.

Polynomials are one of the foundations of algebra, characterized by an algebraic expression consisting of variables, coefficients, and the operation of addition, subtraction, multiplication, and non-negative integer exponents. A polynomial may have one or more terms, such as \(3x^2 - 4x + 15\), which has three terms.

The significance of polynomials comes from their versatility in representing a wide range of problems in mathematics and applied sciences. In the original exercise, the goal is to simplify a polynomial expression which involves not just binomials but also a term with three variables multiplied together, making it a more complex example of polynomial manipulation. Understanding how to simplify such expressions using the distributive property and combining like terms eases the way towards mastering algebraic problems.