Question

Given that (x - 1) is a factor of x4 + 2x3 - 2x - 1, what are the solutions of x4 + 2x3 - 2x - 1 = 0? A. -2, 0, 1 B. -1, 0, 1 C. -2, 1 D. -1, 1

Answer 1

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)