Question

Question #da36a

Answer 1

`y = x^2-x+2-((2x+1)/(x^2+x+1))`

Explanation:

Terms:
Divisor: Generally, this is the denominator. It is the one we are dividing by.
Dividend: Numerator or the one that is getting divided.

First, we will write out the problem in long division form and let us pretend that the square root is the long division symbol:
`(x^2+x+1)sqrt(x^4+0x^3+2x^2-x+1)`
Notice that I included the excluded `x^3` as `0x^3`, this will help you keep track of variables.

We will compare the largest exponent in the divisor to dividend's exponent. The divisor's largest exponent is 2 and the dividend's is 4.

First, look at `x^2` and `x^4`. How many times does `x^2` need to be multiplied to get `x^4`? Twice, so what we will get is x^2 which we will place on top of the division symbol:
`x^2`
`(x^2+x+1)sqrt(x^4+0x^3+2x^2-x+1)`

Then we will multiply our result, `x^2` by the divisor, which we will get `x^4+x^3+x^2`. We will subtract this from the dividend, so what we'll get is:
`x^2`
`(x^2+x+1)sqrt(-x^3+x^2-x+1)`

Then we perform the same action again, however, this time the dividend is `x^3`. Now, remember when you multiply exponents, you add them and if you divide, you minus. So what I said before for `x^2` and `x^4` can be also said as `x^(4-2) = x^2`.

So if you perform the action: `x^3/x^2` or `x^(3-2)` you will get `x`.
Let's add this to `x^2` that we got before:
`x^2+x`
`(x^2+x+1)sqrt(x^4+0x^3+2x^2-x+1)`

Then we multiply x by the divisor to get `-x^3-x^2-x`.

We will then subtract this from the dividend and we will get:
`x^2+x`
`(x^2+x+1)sqrt(2x^2+0x+1)`

Now, we will find out what multiplied by `x^2` gets us `2x^2`, which is a constant 2. Add that to the results:
`x^2+x+2`
`(x^2+x+1)sqrt(2x^2+0x+1)`

Perform the same action by multiplying 2 by the divisor to get `2x^2+2x+2` and subtract from the dividend to get `-2x-1`.

Since our highest exponent for the dividend is smaller than the divisor, we will make a fraction: `(-2x-1)/(x^2+x+1)` Add this to our previous results to get the answer:
`x^2+x+2-(2x+1)/(x^2+x+1)`