Factor each polynomial into linear factors with complex coefficients.

Question

Factor each polynomial into linear factors with complex coefficients.

f(x)=x^3-27i

f(x)=x^4+24

Answer 1

f(x) = x^4 +24

use algebraic identity a^2- b^2 = (a+b)(a-b) where a=x^2 and b=sqr24=2sqr6

a^2+b^2 = a^2 - -b^2 where a=x^2 and -b^2 =-24, sqr-24 = i2sqr6

then use the algebraic identity a 2nd time

f(x) = (x^2+2sqr6)(x^2-2sqr6)

= (x^2+2sqr6)(x+sqr(2sqr6)))(x-sqr(2sqr6))

f(x)= (x+sqr(2isqr6))(x-sqr(2isqr6))(x+sqr(2sqr6))(x-sqr(2sqr6))

f(x) factored into 4 linear factors, 2 imaginary factors and 2 irrational factors, 4 complex factors, all linear