Question

Divide the polynomials by using long division, (x3-x2-2x+14)÷(x-1)

Answer 1

The quotient is x²-2 +12/(x-1).

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Comment on the attachment

You see that I did not maintain strict alignment between quotient terms and dividend terms. For polynomial long division, this is less important than for numerical long division.

In numerical long division, the value of a digit is determined by its place relative to the decimal point and other digits. If a digit gets moved a position right or left, its value changes, so it is important to put and keep digits in the appropriate place. Zeros are used as necessary in the quotient to help do this.

In polynomial long division, each "digit" is the coefficient of some power of the variable. That power is expressed explicitly. We do not depend on the position of the number in the answer to tell us the power of the variable it goes with. (That's slightly different for synthetic division.)

Here, the coefficient of x in the quotient is 0, but no "placeholder" is needed to show that.