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A perfect square trinomial is a specific type of quadratic polynomial, or trinomial, that can be factored into the square of a binomial. In simpler terms, it is a polynomial that takes the form \((m)^2 + 2mn + (n)^2\) or \((m)^2 - 2mn + (n)^2\), leading to the factorized form \((m+n)^2\) or \((m-n)^2\), respectively. These expressions occur when:

• The first term is a perfect square, \(a = m^2\).
• The last term is also a perfect square, \(c = n^2\).
• The middle term should be twice the product of the roots of the perfect squares, either \(2mn\) or \(-2mn\) depending on the sign.

For the polynomial \(36x^2 - 60x + 25\), we found:

• The first term \(36x^2\) is \((6x)^2\).
• The last term \(25\) is \(5^2\).
• The middle term \(-60x\) equals \(-2 \times 6 \times 5x\).

This shows that our polynomial conforms to the perfect square trinomial format, specifically \((6x - 5)^2\). Understanding these conditions helps us factor polynomials effortlessly into their squared forms.

Quadratic polynomials are algebraic expressions of degree two, typically presented as \(ax^2 + bx + c\). The degree of the polynomial refers to the highest power of the variable present, making quadratics distinct due to their second-degree nature. They are ubiquitous in mathematics and serve as the foundation of many algebraic concepts. In the case of our example, \(36x^2 - 60x + 25\),

• \(a = 36\)
• \(b = -60\)
• \(c = 25\)

Quadratics can be factored or solved to find their roots using various methods such as factoring, completing the square, or utilizing the quadratic formula. Factoring is often the first approach when polynomials like our example appear to match well-known forms, like perfect squares. Recognizing such patterns is key in simplifying these expressions.

Binomial expansion refers to the process of expanding expressions that are raised to powers, specifically binomials—expressions containing two terms. A well-known example involves using the formula for squaring a binomial:

• \[(a + b)^2 = a^2 + 2ab + b^2\]

Similarly, the formula applies as:

• \[(a - b)^2 = a^2 - 2ab + b^2\]

These formulas are particularly useful when working to factor quadratics or verify perfect square trinomials. In our specific scenario, the expansion of \((6x - 5)^2\):

• \((6x)^2 = 36x^2\)
• \(-2 \times 6x \times 5 = -60x\)
• \((5)^2 = 25\)

This expansion confirms that \((6x - 5)^2\) equals our original polynomial, validating both the factorization and the original polynomial's identity as a perfect square trinomial.