Question

pls solve this question in picture below

Answer 1

Explanation:

First, we will find the first zero through trial and error.

Let the polynomial be f(x).

When we substitute x with -1, we see that f(-1)=0.

$\mathsf{f(-1)=2x^4+x^3-5x^2-2x+2}$

$\mathsf{=2(-1)^4+(-1)^3-5(-1)^2-2(-1)+2}$

$\mathsf{=2\times1-1-5\times1+2+2}$

$\mathsf{=2-1-5+2+2}$

$\mathsf{=0}$

∴ $\textsf{(x+1) must be a factor of f(x).}$

∴ $\mathsf{f(x)=2x^4+x^3-5x^2-2x+2}$

$\mathsf{=2x^3(x+1)-x^2(x+1)-4x(x+1)+2(x+1)}$

$\mathsf{=(x+1)(2x^3-x^2-4x+2)}$

$\mathsf{=(x+1)[x^2(2x+1)-2(2x+1)]}$

=$\mathsf{(x+1)(2x+1)(x^2-2)}$

$\mathsf{=(x-1)(2x+1)(x+\sqrt2)(x-\sqrt2)}$

∴ The roots/zeroes of the polynomial are 1, -1/2, -√2, √2.

Answer: 1, -1/2, -√2, √2

I hope this helps. :D