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A perfect square trinomial is a special quadratic expression, which arises when a binomial is squared. It takes the form \( ax^2 + bx + c \) and can be recognized when both \( a \) and \( c \) are perfect squares and are combined in the middle by a term \( bx \).
For instance, in our example, the trinomial is \( 9x^2 + 48x + 64 \).
Both 9 and 64 are perfect squares, i.e., \( 9 = 3^2 \) and \( 64 = 8^2 \). The middle term \( 48x \) is twice the product of the square roots of 9 and 64, which confirms the identity of a perfect square trinomial.
This means we can rewrite the trinomial as the square of a binomial \((3x + 8)^2\). Recognizing perfect square trinomials can simplify the process of factoring quadratics significantly.

A quadratic equation is a polynomial equation of degree two. It typically appears in the standard form: \( ax^2 + bx + c = 0 \). In the context of factoring, we often start with the quadratic expression without setting it equal to zero.
The coefficients \( a, b, \) and \( c \) influence the shape and position of the parabola that represents the quadratic on a graph.
In particular, when analyzing a quadratic for factoring, it helps to identify potential perfect square trinomials, like in the example \( 9x^2 + 48x + 64 \), because it can streamline the factorization process. Factoring transforms the quadratic into a product of binomials, making it easier to solve or simplify further.

Polynomial factoring involves breaking down a complex expression into simpler terms or factors that, when multiplied together, produce the original polynomial. This technique is essential for solving quadratic equations, simplifying expressions, and understanding mathematical relations.
In the given exercise, we factor the quadratic \( 9x^2 + 48x + 64 \) by recognizing it as a perfect square trinomial. This helps us rewrite it as \((3x + 8)^2\).
Understanding the properties of numbers and recognizing patterns, such as perfect squares, are crucial skills in polynomial factoring.

• Look for common factors.
• Address special patterns like difference of squares or perfect square trinomials.
• Apply the appropriate factoring method for the situation.

Factoring is an invaluable tool not only in algebra but across various areas of mathematics.