Question

Factor x^10+5 using definition of the sum of squares.

A: (x^2 + sqrt 5i) (x^2 + sqrt 5i)
B: (x^5 - sqrt 5i) (x^5 - sqrt 5i)
C: (x^5 + sqrt 5i) (x^5 - sqrt 5i)
D: (x^5 + sqrt 5i) (x^5 + sqrt 5i)

Answer 1

Recall the difference of squares identity,

a² - b² = (a - b) (a + b)

Let a = x⁵ and b = √5 i, so that a² = x¹⁰ and b² = (√5 i)² = 5 (-1) = -5. Then

x¹⁰ + 5 = x¹⁰ - (-5)

… = (x⁵)² - (√5 i)²

… = (x⁵ - √5 i) (x⁵ + √5 i) … … … [C]