In long division, you divide the usual way. You subtract the whatsamacallit from the old dividend to get a new dividend.

In synthetic division, you "divide" "the reverse" way. You add the "whatsamacallit" with the old dividend to get a new dividend.

Also, in synthetic division, you deal with the numerical coefficients only, while in the long division, you deal with both numerical coefficients and the variables.

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(x^3 +3) / (x -2) == x^3 /x == x^2 first.

Then, (x^2) *(x -2) = x^3 -2x

That "x^3 -2x" is what I call whatsamacallit, because I don't know what am I going to call it.

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Which one is better to play with?

For me, the synthetic division, because it is "faster".

But I am more comfortable with the long division, because it is more reliable or I commit less or no mistakes in using that. And in long division, there is no guessing for the next, new divisor. No trial and error to get going.

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x^3 − 8x^2 + x + 2 by x − 7.

a) (x-7) | x^3 -8x^2 +x +2 === x^3 /x === x^2 ----first quotient.

b) (x^2)*(x-7) = x^3 -7x^2 ------------------------whatsamacallit.

c) (x^3 -8x^2 +x +2) -(x^3 -7x^2) = -x^2 +x +2 ----new dividend.

d) (-x^2) / x === -x

So, new quotient is (x^2 -x)

e) (-x)*(x-7) = -x^2 +7x --------------------------whatsamacallit again.

f) (-x^2 +x +2) -(-x^2 +7x) = -6x +2 ---------------newer dividend.

g)(-6x) / x === -6

So, newer quotient is (x^2 -x -6)

h) (-6)*(x-7) = -6x +42 ----------------you're right, whatsamacallit again.

i) (-6x +2) -(-6x +42) = -40 -----------the remainder.

Therefore,

(x^3 -8x^2 +x +2) / (x-7) = x^2 -x -6, remainder (-40). -----answer.