Answer:
[tex]x = 2 \: or \: x = - 2 \: or \: x = - \frac{1}{2} [/tex]
Step-by-step explanation:
The given polynomial equation is
[tex]2 {x}^{3} + {x}^{2} - 8x - 4 = 0[/tex]
We perform the synthetic division as shown in the attachment by dividing by x-2.
This gives a remainder of 0 and a quotient of
[tex]2 {x}^{2} + 5x + 2[/tex]
This means the polynomial equation becomes:
[tex](x - 2)(2 {x}^{2} + 5x + 2) = 0[/tex]
We factor the quadratic term by splitting the middle term;
[tex](x - 2)(2 {x}^{2} + 4x +x + 2) = 0[/tex]
[tex](x - 2)(2 x(x+ 2) +1(x + 2) )= 0[/tex]
Collect common factors again:
[tex](x - 2)((x+ 2)(2x + 1) = 0[/tex]
The solution is:
[tex]x = 2 \: or \: x = - 2 \: or \: x = - \frac{1}{2} [/tex]
