Use long division to divide
x^3 − 8x^2 + x + 2 by x − 7.
Also, what is the difference between synthetic and long division?
Which of the two is best to play with?
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.