Factor each polynomial. See Examples 5 and 6. (p-2q)^2-100

Verified step by step guidance

1. Recognize the expression \((p-2q)^2 - 100\) as a difference of squares.
2. Recall the difference of squares formula: \(a^2 - b^2 = (a-b)(a+b)\).
3. Identify \(a = (p-2q)\) and \(b = 10\) since \(100 = 10^2\).
4. Apply the difference of squares formula: \((p-2q)^2 - 10^2 = ((p-2q) - 10)((p-2q) + 10)\).
5. Write the factored form: \((p-2q-10)(p-2q+10)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for simplifying expressions, solving equations, and understanding the polynomial's roots. Common techniques include identifying common factors, using the difference of squares, and applying the quadratic formula when necessary.

Difference of Squares

The difference of squares is a specific factoring technique used when a polynomial is in the form a^2 - b^2. It can be factored into (a - b)(a + b). In the given polynomial, (p-2q)^2 - 100 can be recognized as a difference of squares, where a = (p-2q) and b = 10, allowing for straightforward factoring.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, such as (a + b)^n. In this context, (p-2q)^2 can be expanded using the formula (a - b)^2 = a^2 - 2ab + b^2. Understanding this concept is crucial for recognizing how to manipulate and factor polynomials effectively.