Question

What must be added to x3+3x2+12x+9 so that the resulting polynomial is completely divided by x2+x-6

Answer 1

x² + x -6 ) x³ - 3x² -12x + 19 ( x -4

x³ + x² -6x

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-4x² -6x + 19

-4x²-4x + 24

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-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)