Precal Question

Question

x^4 - 25

find all zeros (real and complex) and factor the polynomial completely. STEP BY STEP FOR ALL PLEASE.

Answer 1

x⁴ - 25 is the difference of two squares. x⁴ is a square (x² • x²) and 25 is a square (5•5) so factors of the difference of these two squares are (x² + 5) and (x² - 5). So:

f(x) = x⁴ - 25 is factored into f(x) = (x² + 5)(x² - 5)

To find the zeros, we set the function equal to zero:

0 = (x² + 5)(x² - 5) to solve, set each piece equal to zero so:

A)

x² + 5 = 0

x² = -5

x = ±√-5

x = ±√5 i

B)

x² - 5 = 0

x² = 5

x = ±√5