Question

x^4+1, Find the complex zeros and write in factored form

Answer 1

do you know the difference of squares?

x^2 - y^2 = ( x - y) ( x + y )

Answer 2

well if you dont, you know it now, makes sense so far?

Answer 3

expand the right hand side
(x-y)(x+y) = x^2 -xy + yx - y^2 = x^2 - y^2

Answer 4

lets look at simpler case
x^2 + 1 = x^2 - i^2 , do you see why? because i^2 = -1

Answer 5

So (x^2-1)(x^2+1) are the complex zeroes? I thought i^2 was 1

Answer 6

i^2 = -1, remember the definition of i, it is the square root of -1

Answer 7

What would the complex zeros be?

Answer 8

well this problem has two steps

Answer 9

lets do a substitution.
let u = x^2

then we need to factor
u^2 + 1

ok so far?

Answer 10

Yes

Answer 11

and how do we factor u^2 + 1 ?

Answer 12

ok it is true that
u^2 + 1 = u^2 - i^2

STOP! do you see why, if not let me know

Answer 13

I think I got that

Answer 14

because
i^2 = -1
and so
u^2 +1 = u^2 - (-1) = u^2 - i^2

Answer 15

Okay I get that now

Answer 16

u^2 + 1 = u^2 - i^2 = (u - i ) ( u + i ) , by difference of squares

Answer 17

Okay got that too

Answer 18

ok now we plug back

(x^2 - i ) ( x^2 + i )

Answer 19

since u = x^2

Answer 20

@perl But that would give me x^4+1 not x^4-1