Precalculus 12 grade

Question

Find the complex zeros of f(x)=x^4+4x^2-32. Use your results to write the factors.

Answer 1

x^4 +4x^2 -32

let y = x^2

y^2 +4y -32

factor, complete the square or use the quadratic formula

(y +8)(y-4)

= (x^2 +8)(x^2-4) = (x^2+8)(x+2)(x-2)

x^2 = -8, x = +/-2isqr2

+/-2isqr2 are the imaginary roots

corresponding factors are

(x +2isqr2) and (x-2isqr2)

2 imaginary roots and 2 real roots

real roots are 2 and -2

Answer 2

f(x)=x^4+4x^2-32

To find zero, make f(x) = 0 and factor the polynomial

x^4+4x^2-32 = 0

(x^2+8)(x^2-4) =0

(x^2+8)(x+2)(x-2)=0

use zero product property to solve

(x^2+8)=0 0

x^2 = -8

x = +√-8, - √-8

x = 2i√2, -2i√2

(x+2) =0

x= -2

(x-2)=0

x=2

Zeros of the functions are 2 real and 2 imaginary

x= -2, 2, 2i√2, -2i√2

Factors of the functions are

f(x) = (x+2i√2)(x-2i√2)(x+2)(x-2)

Answer 3

f(x) = x⁴ + 4x² - 32 = 0

(x² + 8)(x² - 4) = 0

x² = -8 or x² = 4

x = ±√(-8) or ±√4

So, x = ±2√2 i or ±2

f(x) = (x + 2)(x - 2)(x - 2√2 i)(x + 2√2 i)