Factor the polynomial P(x) = x^4 − 16x^2 − 225 completely into linear factors with complex coefficients

Question

The answer is (x + 3i)(x − 3i)(x + 5)(x − 5) but I really don't understand why it isn't (x + 5i)(x − 5i)(x + 3)(x − 3)

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Also, what do you think of this answer:_

P(x) = x^4 − 16x^2 − 225

Let u = x^2

u^2 - 16u -255

(u+9)(u-25)

(x^2 +9)(x^2 -25)

*applying the Quadratic formula on (x^2 +9)*

A=0

B=-9i

*applying the Quadratic formula on (x^2 -25)*

A=25!

B=0

so the answer is weather

(x + 3i)(x − 3i)(x + 5)(x − 5)

or

(x + 5i)(x − 5i)(x + 3)(x − 3)

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btw I really dont understand the Quadratic formula purpose at all, but I think It's used to find the complex coefficients. Is that right?

Answer 1

x⁴ - 16x² - 225

Let u = x²

u² - 16u - 225

Factors of -225 that add to -16 are (-25)(9)

(u-25)(u+9)

Replace u with x²

(x²-25)(x² + 9)

 

x² -25 is the difference of 2 squares

a²-b² = (a-b)(a+b)

x²- 25 = (x-5)(x+5)

 

The quadratic can be used on x² + 9 to find the roots

x² + 0x + 9 =0 ...,    a=1, b=0, c=9

x = ±√[(-4)(9)]/2 = ±[√(-36)]/2 = ±6i/2 = ±3i

 

x² + 9 = (x+3i)(x-3i)

 

x⁴ - 16x² - 225 = (x-5)(x+5)(x-3i)(x+3i)