Question

question 12
0.84 pt
what is the factored form of the expression over the complex numbers?
4x² + 25y²
○ (2x − 5iy)(2x − 5iy)
○ (2x + 5y)(2x + 5y)
○ (2x + 5iy)(2x + 5iy)
○ (2x + 5iy)(2x − 5iy)

Explanation:

Step1: Recall the difference of squares formula

The difference of squares formula is \(a^2 - b^2=(a + b)(a - b)\). We can rewrite the given expression \(4x^{2}+25y^{2}\) as a difference of squares by using the fact that \(i^{2}=- 1\), so \(25y^{2}=-(- 25y^{2})=- (5iy)^{2}\) and \(4x^{2}=(2x)^{2}\). Then the expression becomes \((2x)^{2}-(5iy)^{2}\).

Step2: Apply the difference of squares formula

Using the formula \(a^{2}-b^{2}=(a + b)(a - b)\) where \(a = 2x\) and \(b=5iy\), we get \((2x + 5iy)(2x-5iy)\).

Answer:

\((2x + 5iy)(2x - 5iy)\) (the fourth option: \((2x + 5iy)(2x - 5iy)\))