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

Verified step by step guidance

1. Recognize that the given expression \( (p-2q)^2 - 100 \) is a difference of squares, since it can be written as \( (p-2q)^2 - 10^2 \).
2. Recall the difference of squares formula: \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = p - 2q \) and \( b = 10 \).
3. Apply the formula by substituting \( a \) and \( b \) into it: \( (p - 2q - 10)(p - 2q + 10) \).
4. Write the factored form clearly as \( (p - 2q - 10)(p - 2q + 10) \).
5. Verify by expanding the factors to ensure they multiply back to the original expression.

Key Concepts

Difference of Squares

The difference of squares is a factoring technique used when an expression is in the form a² - b². It factors into (a - b)(a + b). Recognizing this pattern helps simplify expressions like (p - 2q)² - 100 by identifying a = (p - 2q) and b = 10.

Binomial Squares

A binomial square is an expression of the form (x ± y)², which expands to x² ± 2xy + y². Understanding how to expand and recognize binomial squares is essential for factoring or simplifying expressions involving squared binomials, such as (p - 2q)².

Polynomial Factoring

Polynomial factoring involves rewriting a polynomial as a product of simpler polynomials. Mastery of various factoring methods, including factoring by grouping, special products, and difference of squares, is crucial for breaking down complex expressions into factors.