Product of linear factors and zeros of the functions

Question

Write the polynomial as the product of linear factors.

g(x) = x⁴ − 2x³ + 5x² − 8x + 4

g(x) =

List all the zeros of the function. (Enter your answers as a comma-separated list.)

x =

This is what I've gotten twice and it's incorrect.

g(x)=(x-1)^2(x^2+4)

x=1,1+\sqrt{2i},\ -\sqrt{2i}

Answer 1

You have correctly factored the polynomial. One of the issues might be with your complex roots. To find them, set x² + 4 = 0. Then, we get x² = -4. Finding the square root of both sides, we see that there are two solutions, a negative and a positive one, as we are taking a square root. The positive solution is x = sqrt(-4), and the negative one is x = -sqrt(-4). In both cases, sqrt(-4) = 2i, so our two solutions are x = 2i and x = -2i.

Answer 2

Your factored form is correct g(x)=(x-1)²(x²+4) but does not go far enough. It should be

(x-1)² (x-2i)(x+2i). The zeros are 1,1,2i, -2i