Use synthetic division to show that the number given to the right of the equation is a solution of the equation. Then solve the polynomial equation. 2x^(3)+x^(2)-8x-4=0; 2

Question

asked Jul 16, 2021 198k views

1 Answer

Leo Brueggeman · Answer 1 · 2021-07-20T08:42:25+0000

Answer:

$x = 2 \: or \: x = - 2 \: or \: x = - (1)/(2)$

Explanation:

The given polynomial equation is

$2 {x}^(3) + {x}^(2) - 8x - 4 = 0$

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

$2 {x}^(2) + 5x + 2$

This means the polynomial equation becomes:

$(x - 2)(2 {x}^(2) + 5x + 2) = 0$

We factor the quadratic term by splitting the middle term;

$(x - 2)(2 {x}^(2) + 4x +x + 2) = 0$

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

Collect common factors again:

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

The solution is:

$x = 2 \: or \: x = - 2 \: or \: x = - (1)/(2)$