Explanation:
x⁴ + 5x³ - 6x² - 5x - 2
should become
x³ - 3x² + x -1
so, what do I need to subtract from the top terms to get the bottom terms ?
because we can only look from one x- term to the next. we can only combine them, when they have the same x exponent.
so, we see there is no x⁴ in the desired result.
therefore, we know as a first item that x⁴ had to be subtracted.
the next x exponent is 3.
we have 5x³ on the top, and x³ on the bottom.
so, clearly, we need to subtract 4x³ as second term.
the next x exponent is 2.
we have -6x² on the top, and -3x² on the bottom.
so, we should actually add 3x². and how do we do this in a subtraction ? by subtracting a negative value, which actual adds it.
so, we need to subtract -3x² as third term.
the next x exponent is 1 (just x).
we have -5x on the top and x on the bottom.
so, the same trick as before, as we need to add something.
so, we need to subtract -6x as fourth term.
and finally, the x exponent 0, which just shows the constant numbers.
we have -2 on the top and -1 on the bottom.
again the same trick to actually add 1.
we need to subtract -1 as fifth term.
so, in total, this is what we need to subtract from the top line to get the bottom line :
x⁴ + 4x³ - 3x² - 6x - 1