g(x) = 2x^5 - x^4 - 5x^3 + 10x^2 - 2x - 4?

Question

Find all the zeros of the following:

g(x) = 2x^5 - x^4 - 5x^3 + 10x^2 - 2x - 4

Answer 1

g(x) = 2x^5 - x^4 - 5x^3 + 10x^2 - 2x - 4

= (2x + 1)(x + 2)(x - 1)(x² - 2x + 2) = 0

x = -2, -1/2, 1

There are three real solutions.

Answer 2

have to use long division of the polynomial by guessing its roots.
the sequence that i've tried out is as below:

1. x = 1 to get G(x) = 0, so x = 1 is a root. divide the polynomial 2x^5 - x^4 - 5x^3 + 10 x^2 - 2x -4 by the expression (x - 1), to get 2x^4 + x^3 - 4x^2 + 6x +4.

2. x = -0.5 to get polynomial from (1) = 0. so divide 2x^4 + x^3 - 4x^2 + 6x +4 by the expression (x + 0.5) to get 2x^3 - 4x + 8.

3. then, x = -2 to get 2x^3 - 4x + 8 = 0. so divide 2x^3 - 4x + 8 by the expression (x + 2) to get 2x^2 - 4x + 4.

4. solve 2x^2 - 4x + 4 = 0 using quadratic method.
get x = 1 + i and x = 1 - i (where i denotes imaginary root).

so u have 5 roots (for a fifth order polynomial), i.e. x = 1, -0.5, -2, 1 + i, and 1 - i.

pls check.

Answer 3

-5x^3+10x^2-4