The Correct Answer and Explanation is:
Correct Answer: (y – 7i)(y + 7i)
Explanation:
The task is to factor the expression y² + 49 over the set of complex numbers. This expression is a specific type known as a “sum of squares”. While a sum of squares cannot be factored using only real numbers, it can be factored using complex numbers by applying the “difference of squares” formula.
The difference of squares formula is a² – b² = (a – b)(a + b). Our goal is to manipulate y² + 49 so it fits this pattern. To achieve this, we introduce the imaginary unit, i, which is defined by the fundamental property i² = -1. This property allows us to convert a positive term into a negative one.
First, we can rewrite the expression y² + 49 as a subtraction:
y² + 49 = y² – (-49)
Next, we need to find a term that, when squared, equals -49. Using the imaginary unit i, we can express -49 as the square of 7i:
(7i)² = 7² * i² = 49 * (-1) = -49
Now we can substitute (7i)² back into our expression:
y² – (-49) = y² – (7i)²
This expression is now in the form of a difference of squares, a² – b², where a = y and b = 7i. We can apply the factoring formula (a – b)(a + b):
y² – (7i)² = (y – 7i)(y + 7i)
Therefore, the expression y² + 49 factored over the complex numbers is (y – 7i)(y + 7i). We can verify this by multiplying the factors, which results in y² + 7iy – 7iy – 49i², simplifying to y² + 49.
