Question

What must be added to x³ - 3x²- 12x + 19 so that the result is exactly divisibly by x² + x - 6?

Answer 1

2x + 5 must be added to x³ - 3x²- 12x + 19 so that the result is exactly divisible by x² + x - 6

Step-by-step explanation:

What must be added to x³ - 3x²- 12x + 19 so that the result is exactly divisible by x² + x - 6

                      x - 4

                      _____________

x² + x - 6    _ | x³ - 3x²- 12x + 19  |_

                        x³ + x²  - 6x

                       _______________

                             -4x²  - 6x + 19

                              -4x² - 4x  + 24

                               ____________

                                        -2x  - 5

Remainder = - 2x - 5 = - (2x + 5)

Remainder must be subtracted from polynomial to get  exactly divisibly by  x² + x - 6

Hence -ve of remainder must be added

= - (-(2x + 5))

= 2x + 5

2x + 5 must be added

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Answer 2

$\huge\star\mathfrak\blue{{Answer:-}}$

X² + x -6 ) x³ - 3x² -12x + 19 ( x -4

x³ + x² -6x

--------------------------------

-4x² -6x + 19

-4x²-4x + 24

-------------------------------------

-2x - 5

hence ,

x³ - 3x² -12x + 19 = (x² +x - 6)(x - 4) -(2x + 5)

 x³ - 3x² -12x + 19 + (2x + 5)  = (x² + x - 6)(x - 4)

hence if we add (2x + 5)