Kuta Software - Infinite Algebra 1 Factoring Special Cases Factor each completely. 1) 16n2−9 3) 16b2
The Correct Answer and Explanation is:
The given expression to factor is 16n² – 9.
Recognizing this as a difference of squares, we apply the formula a² – b² = (a + b)(a – b). In this case, 16n² is a perfect square, written as (4n)², and 9 is also a perfect square, written as 3². Using the formula, we factor the expression as follows:
16n² – 9 = (4n + 3)(4n – 3)
Explanation:
To solve this problem, we first identify whether the given polynomial fits a recognizable factoring pattern. The expression 16n² – 9 consists of two terms, and both are perfect squares. The first term, 16n², can be rewritten as (4n)², while the second term, 9, is rewritten as 3². Since the expression is written in the form a² – b², we apply the difference of squares formula, which states that a² – b² factors into (a + b)(a – b).
We proceed by setting a = 4n and b = 3. Substituting these values into the formula, we obtain:
(4n)² – 3² = (4n + 3)(4n – 3)
This method simplifies the given quadratic expression into a product of two binomials. Factoring through the difference of squares is useful because it allows algebraic expressions to be broken down into simpler multiplicative components, making further calculations and problem-solving easier. In mathematics, recognizing special factoring cases like the difference of squares helps in solving equations efficiently without resorting to complex methods such as the quadratic formula. The difference of squares appears frequently in algebra and higher-level mathematics, making it an essential skill for polynomial manipulation and simplification.
Thus, the completely factored form of the given expression is (4n + 3)(4n – 3).
