Hello markYou need to use algebraic long division. There's an example here.
See if you can apply this to your example. You should get a quotient and a remainder . Can you then see what to do with the remainder? (Think about how you handle the remainder when you divide using ordinary numbers.
Grandad
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without latex, ugly result
11111 __x_________
x + 2 / x^2 + 3x + 1
000000x^2 + 2x (subtract)
000000000000x + 1 ( bringdown 1)
then next term will be 1
00000 x + 1 0
x + 2 / x^2 + 3x + 1
000000x^2 + 2x (subtract)
000000000000x + 1 ( bringdown 1)
000000000000x + 2 (subtract)
00000000000000 - 1 ( remainder)
thus =
Mathaddict method is even beautiful,
, how?
x^2 + 3x + 1 = x^2 + 3x + (1 + 2 - 2), you breakdown the 3rd-term
then, rearrange (x^2 + 3x + 2) - 1, you will get
(x + 1)(x + 2) - 1
divide by (x + 2) each term
(x + 1)(x + 2)/(x + 2) - 1/(x + 2), notice the (x + 2) term cancels
thus, (x^2 + 3x + 1)/(x + 2) = (x + 1) - 1/(x + 2).
easy right?