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The term 'quadratic trinomial' describes a polynomial of degree 2 that has three terms. It generally follows the format:

• \(ax^2 + bx + c\)

Here, 'a', 'b', and 'c' represent constants, while 'x' is the variable. Quadratic trinomials can often be factored into simpler, multiplied expressions. Factoring is crucial as it helps in solving quadratic equations and simplifying algebraic expressions. In this specific example \(p^2 - 10pq + 25q^2\), 'p' and 'q' are variables and the polynomial fits the quadratic trinomial structure.
At first glance, \(p^2\) matches \(a^2x^2\), \(-10pq\) matches \(b\), and \(25q^2\) matches \(c\), fitting the general format of a quadratic trinomial.

A perfect square trinomial is a special type of quadratic trinomial which can be written as the square of a binomial expression. The structure of a perfect square trinomial is:

• \(a^2 - 2ab + b^2 = (a - b)^2\)

Understanding this form aids in identifying and factoring these trinomials. In our exercise, \(p^2 - 10pq + 25q^2\) needs to be checked if it fits this structure. Let’s identify the respective components here:

• \(a^2 = p^2\) means \(a = p\)
• \(-2ab = -10pq\) means \(-2 \times p \times b = -10pq\), solving for 'b', yields \(b = 5q\)
• \(b^2 = 25q^2\) confirms \(b = 5q\)

All the conditions are verified to fit the form of a perfect square trinomial. Therefore, \(p^2 - 10pq + 25q^2 = (p - 5q)^2\).

The factored form of a polynomial is the expression rewritten as the product of its factors. It simplifies understanding the roots and solving equations. In our specific problem, since we verified that \(p^2 - 10pq + 25q^2\) is a perfect square trinomial, it can be expressed in its factored form. Therefore: \[(p - 5q)^2\]
Expressing the original trinomial this way highlights the factors and eases any further problem-solving or analysis. By rewriting as a binomial square, we also immediately see that \( (p - 5q)\) is a repeated factor. Thus, the factored form is \( (p - 5q)\) multiplied by itself, or \( (p - 5q)^2\). This outcome demonstrates practical ways quadratic trinomials interact with concepts like factoring and solving quadratic equations.