SOLUTION: 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)

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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