In Exercises 15–58, find each product. (x+5)(x−5)

Verified step by step guidance

1. Recognize that the expression $(x+5)(x-5)$ is a difference of squares.
2. Recall the formula for the difference of squares: $(a+b)(a-b)=a^2-b^2$.
3. Identify $a$ and $b$ in the expression: here, $a=x$ and $b=5$.
4. Apply the difference of squares formula: substitute $a$ and $b$ into $a^2-b^2$.
5. Simplify the expression to get the final result.

Key Concepts

Factoring and Difference of Squares

The expression (x+5)(x−5) is an example of the difference of squares, which follows the formula a² - b² = (a + b)(a - b). In this case, a is x and b is 5. Recognizing this pattern allows for quick simplification of the product without expanding it fully.

Algebraic Expansion

Algebraic expansion involves multiplying two binomials to obtain a polynomial. For (x+5)(x−5), you can apply the distributive property (also known as the FOIL method) to find the product, resulting in x² - 25. Understanding how to expand binomials is crucial for solving polynomial expressions.

Polynomial Simplification

After expanding the product of the binomials, the next step is polynomial simplification, which involves combining like terms and reducing the expression to its simplest form. In this case, the result x² - 25 is already simplified, illustrating the importance of recognizing when an expression cannot be reduced further.