Question 955352: Hello, have a good day. I need help here with this polynomial division. Could someone explain its solution to me step by step? Thank you.
"Divide (x^4)+(x^2)-(x)+(1) by (x^2)+(2) and check."
Thank you in advance.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! you want to divide x^4 + x^2 - x + 1 by x^2 + 2
you should fill in the dividend with the missing orders.
your dividend becomes x^4 + 0x^3 + x^2 - x + 1
divide x^4 by x^2 to get x^2.
multiply x^2 + 2 by x^2 to get x^4 + 2x^2
subtract x^4 + 2x^2 from x^4 + 0x^3 + x^2 - x + 1 to get:
0x^3 - x^2 - x + 1
the 0x^3 cancels out of the equation because it has no value and you are left with:
-x^2 - x + 1
divide -x^2 by x^2 to get -1.
multiply x^2 + 2 by -1 to get -x^2 - 2
subtract -x^2 - 2 from -x^2 - x + 1 to get:
-x + 3
since the leading term is less than x^2, you are done and this is your remainder.
your solution is that x^4 + x^2 - x + 1 divided by x^2 + 2 is equal to x^2 - 1 with a remainder of -x + 3.
to confirm, you want to multiply x^2 + 2 by x^2 - 1 and then you want to add the remainder of -x + 3 to the result.
(x^2 + 2) * (x^2 - 1) is equal to x^4 - x^2 + 2x^2 - 2
simplify to get x^4 + x^2 - 2
add -x + 3 to this to get:
x^4 + x^2 - x + 1
since that's the same as your original equation, you did good.
the manual calculations by hand are shown below:
note that, in the division, where i show -x^4 - 2x^2, that's really - (x^4 + 2x^2).
you are subtracting x^4 + 2x^2 which is the same as adding -x^4 - 2x^4.
also, in the division, where i show + x^2 + 2, that's really - (-x^2 - 2).
you are subtracting -x^2 - 2 which is the same adding x^2 + 2.
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