91Ó°ÊÓ

The difference of squares is a special type of polynomial factorization. It applies when you have two perfect squares separated by a subtraction sign. The general form of a difference of squares is: \[a^2 - b^2 = (a + b)(a - b)\]In this case, we have: \[16 - (x + 3y)^2\]Here, 16 is the square of 4, and \[(x + 3y)^2\] is already a perfect square. Thus, we can rewrite the expression as \[ (4)^2 - (x + 3y)^2\]This matches the difference of squares pattern perfectly. From here, we can apply the difference of squares formula.

A perfect square is a term that results from squaring a binomial. For example, \[ (x + 3y)^2\] means \[ (x + 3y) * (x + 3y)\].Recognizing perfect squares is essential for identifying when you can apply the difference of squares formula. In our problem, \[ 16\] is a perfect square because it is \[ 4^2\]. Similarly, \[ (x+3y)^2\] suggests that \[ x+3y\] is the binomial being squared. Recognizing both 16 and \[ (x+3y)^2\] as perfect squares allows us to factor the polynomial efficiently.

Polynomial factorization is the process of breaking down a polynomial into a product of simpler polynomials. By factoring, you express the polynomial as the product of its factors. For our given exercise, we start by identifying the polynomial:\[16 - (x + 3 y)^2\]. Next, we recognize it fits the difference of squares form, leading us to the expression: \[ (4)^2 - (x + 3y)^2\]. Using the difference of squares formula: \[a^2 - b^2 = (a + b)(a - b)\], where a = 4 and b = (x + 3y), we obtain the factored form: \[ (4 + (x + 3y))(4 - (x + 3y))\].' Finally, simplifying the expressions inside the parentheses, we get: \[(4 + x + 3y)(4 - x - 3y)\].This fully factored form allows easier analysis and solving of the polynomial, showcasing the importance of mastering factorization techniques.