Question

What must be separated from x power 4 + 2 x power 3-13 x square -12x + 21 such that the resulting polynomial is exactly divisible by X square + 2 X + 3

Answer 1

Given polynomial :

x⁴ + 2 x³ - 13 x² - 12 x + 21 .

Let us say that k is separated from it.

Then the resulting polynomial will be :

x⁴ + 2 x³ - 13 x² - 12 x + 21  - k

IF YOU STRICTLY GO :

The value of k cannot be determined as it is complex .

But if the question is like this :

Let f ( x ) = x⁴ + 2 x³ - 13 x² - 12 x + 21 .

This is divisible be x² + 2 x + 3

Hence we can say x² + 2 x + 3 is a factor of f ( x )

I think the question will be x² + 2 x - 3 because otherwise the solution has NO real roots.

So the question if was like that :

x² + 2 x - 3 = 0

= x² + 3 x - x - 3 = 0

= x ( x + 3 ) - 1 ( x + 3 ) = 0

= ( x + 3 )( x - 1 ) = 0

x = - 3 or x = 1

By factor theorem :

If you put this in f (x) we get 2 values of k .

( 1 )⁴ + 2 ( 1 )⁴ - 13 ( 1 )² - 12 ( 1 ) + 21 - k = 0

= 1 + 2 - 13 - 12 + 21 - k = 0

= - 1 - k = 0

= k = 1

Or

( - 3 )⁴ + 2 ( - 3 )³ - 13 ( - 3 )² - 12 ( - 3 ) + 21

= 81 - 54 - 117 + 36 + 21

= 90 + 36 + 21

= 126 + 21

= 127

So either 127 or 1 is the answer .

Answer 2

hey mate,here is ur answer

Given : Dividend =

Divisor =

Solution :

Since we know that :

Since the remainder is  2x-3

So, 2x-3 must be subtracted from    so that the resulting  polynomial is exactly divisible

2×-3=-1

hope it helps u plz mark as brainlist