Find all rational zeros of the polynomial Enter your answers

Find all rational zeros of the polynomial. (Enter your answers as a comma-separated list.)

P(x) = x⁴ 8x³ + 12x² + 32x 64

x =  

Determine any irrational zeros. (Enter your answers as a comma-separated list. If there are no irrational zeros, enter NONE.)
x =  

Factor the polynomial completely.
P(x) =  

Solution

P(x) = x^4 8x^3 + 12x^2 + 32x 64

= (x^4 + 12x^2 64) + (-8x^3 + 32x)

let: x^2 = a

P(x) = (a^2 + 12a - 64) - 8x(x^2 - 4)

= (a + 16)(a - 4) - 8x(x^2 - 4)

=  (x^2 + 16)(x^2 - 4) - 8x(x^2 - 4)

= (x^2 - 4)[x^2 - 8x + 16]

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

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

Set P(x) = 0

x = 2, -2, 4, 4