Question

PLEASE HELP ASAP

Answer 1

Hey there!!

Using the remainder theorem :

x + 1 = 0

x = -1

The remainder is -12.

Plugging in the values :

... 2x³ - mx² + nx - 2 = -12

... 2x³ - mx² + nx = - 10

... 2 ( - 1 )³ -m ( -1 )² + n ( -1 )

... -2 - m -  n = - 10

... - m - n = -8

Let's take the other one :

x - 2 = 0

x = 2

Plugging in the values :

2x³ - mx² + nx - 2 = 0

2x³ - mx² + nx = 2

2 ( 2 )³ - m ( 2 )² + n ( 2 ) = 2

16 - 4m + 2n = 2

- 4m + 2n = -14

Let's get both the equations together :

- m - n = -8  ---- ( 1 )

- 4m + 2n = -14  ---- ( 2 )

......................................................................................................................................................................

Multiply the first equation with 2

- 2m - 2n = -16 --- ( 1 )

- 4m + 2n = -16 --- ( 2 )

Add both the equations.

...................................................................................................................................................................... - 6m = - 32

m = -32 / - 6

m = 32 / 6

m = 16 / 3

Substitute this value to get the value of ' n '

- m - n = - 8

- 16 / 3 - n = - 8

- n = - 8 + 16 / 3

- n = - 8 / 3

n = 8 / 3

m = 16 / 3 and n = 8 / 3

Hope my answer helps!!

Answer 2

m = 5 and n = 3

let f(x) = 2x³ - mx² + nx - 2 , then

f( - 1 ) = - 2 - m - n - 2 = - 12

f(2 ) = 16 - 4m + 2n - 2 = 0

Tidying up the 2 equations

- m - n = - 8 → (1)

- 4m + 2n = - 14 → ( 2)

multiply equation ( 1) by 2

- 2m - 2n = - 16 → ( 3 )

add ( 2 ) and ( 3 ) term by term

-6m = - 30 ⇒ m = 5 ( substitute m = 5 into (2 )

- 20 + 2n = - 14

2n = 6 ⇒ n = 3