Question

Without actual division prove that (x^2-x-2) divides
2x^4 + x^ 3- 5x² - 8x-4.
please tell step by step ​

Answer 1

Answer:

it is divisible by ( x^2-x-2)

i.e.

$x^2 - x - 2$

can be written as

$x^2 - 2x + x - 2$

$x(x - 2) + 1(x - 2)$

$(x - 2)(x + 1)$

we got two results

1. (x-2)
2. (X+1)

$(x - 2) =0 \\ x = 2$

put X=2 in first equation i.e.

$2x ^4 + x^3 - 5x - 8x - 4$

putting X = 2

we will get 0 as answer means it is divisible by

X= 2

now,

$(x + 1) = 0 \\ x = - 1$

put X= -1 in the equation

$2x ^4 + x^3 - 5x ^2 - 8x - 4$

putting X= -1

we will get 0 as answer then it is also divisible by X= -1

conclusion ....

we got this X=2 and X= -1 from equation

$x^{2} - x - 2$

so it means this equation divides the the equation

$2x^4 + {x}^{3} - {5x}^{2} - 8x - 4$