Answer:
D. -1, 1
Step-by-step explanation:
If x-1 is a factor of x⁴ + 2x³ - 2x - 1, then x-1 =0 is a factor
if x-1 = 0
x = 1
To get other factors we will divide the polynomial by x-1
Find the division in the attachment
x⁴ + 2x³ - 2x - 1,/x-1 = x³ +3x²+ 3x + 1,
let x = -1 be a factor of the resulting expression
f(-1) = (-1)³ +3(-1)²+ 3(-1) + 1
f(-1) = -1 +3(1)-3 + 1
f(-1) =-1+1-3+3
f(-1) = 0
Since f(-1) =0, hence x+1 is a factor of the resulting polynomial'
Dividing x³ +3x²+ 3x + 1, by x+1 to reduce the power of the polynomial
x³ +3x²+ 3x + 1/x+1 = x²+2x+1
Factorizing x²+2x+1
= x²+2x+1
=x²+x+x+1
= x(x+1)+1(x+1)
= (x+1)(x+1)
if the function is equal to zero
x+1 = 0 and x+1 = 0
x = -1 twice
Hence the solutions to the polynomial are 1 and -1(three times)
