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The difference of squares is a special algebraic expression that can be factored using a specific formula. This expression takes the form \[A^2 - B^2\] and represents the subtraction of two squared terms. It's called the 'difference of squares' because it involves two squares being subtracted. To factor a difference of squares:

• Identify the terms \(A\) and \(B\) such that the expression fits the form \(A^2 - B^2\)
• Apply the formula: \(A^2 - B^2 = (A + B)(A - B)\)
• Substitute the identified terms back into the formula

This method simplifies the factoring process and reveals the factors of the original expression. In the exercise provided, we have the expression \[64a^2 - 25\]. We recognize this as a difference of squares with \(A = 8a\) and \(B = 5\). Applying our formula, we factor it into: \[(8a + 5)(8a - 5)\].

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations like addition, subtraction, multiplication, and division. In the context of factoring, understanding the structure of algebraic expressions is crucial. The given expression, \[64a^2 - 25\], is a polynomial, which is a special type of algebraic expression consisting of terms added or subtracted together. Each term in a polynomial is a product of a constant coefficient and one or more variables raised to non-negative integer powers. To factor such expressions effectively, we often use special formulas and techniques. Recognizing patterns, such as the difference of squares (like in this case), allows us to simplify and break down complex expressions into simpler, factorable parts.

Factoring formulas are key tools in breaking down complex algebraic expressions into their constituent parts. One common formula is the difference of squares, \[A^2 - B^2 = (A + B)(A - B)\], which we used in the exercise. Here are a few more key factoring formulas every student should know:

• Perfect square trinomial: \(A^2 ± 2AB + B^2 = (A ± B)^2\)
• Sum and difference of cubes: \(A^3 + B^3 = (A + B)(A^2 - AB + B^2)\) and \(A^3 - B^3 = (A - B)(A^2 + AB + B^2)\)
• General trinomial: \(Ax^2 + Bx + C\) can be factored into the product of two binomials when possible

Factoring simplifies expressions and is useful in solving equations, understanding polynomial identities, and simplifying fractions. Make sure to familiarize yourself with these formulas and practice applying them to various expressions.