Question

Factor the expression completely I’ve the complex numbers X^4+10x^2+25

Answer 1

We have

(a + b)² = a² + 2ab + b²

so that with a = x² and b = 5, we have

x⁴ + 10x² + 25 = (x² + 5)²

Next, we have

a² - b² = (a - b) (a + b)

so that with a = x and b = √5 i,

x² + 5 = x² - (-5) = (x - √5 i) (x + √5 i)

So, the complete factorization over the complexes is

(x - √5 i)² (x + √5 i)²