I'm not following logic of using long division on polynomials.
If you are using regular long division, we would do the following:
+----------
2 | 86
How many times does 2 go into 8? 4. Multiply 2 by 4 and drop down the next tenth power. So we end up with 6.
4
+----------
2 | 86
8
-----------
6
How many times does 2 go into 6. 3. So we add that to the next tenth place, and drop it down and continue doing this until we get 0 or a remainder.
43
+----------
2 | 86
8
-----------
6
6
-----------
0
Now I am asked to do the same thing but with polynomials.
+----------
x + 3 | x^2 + 10x + 21
In order to solve this, we are told to determine how many times does x go into x^2. The answer is x because x*x=x^2. Then we are told to multiply that x by the whole divisor x + 3, so we get a result that looks like this:
x
+---------------
x + 3 | x^2 + 10x + 21
x^2 + 3x
----------------
7x + 21
And then again we ONLY check how many times x goes into 7x, which is 7 times, so our result looks like this:
x + 7
+---------------
x + 3 | x^2 + 10x + 21
x^2 + 3x
----------------
7x + 21
7x + 21
----------------
0
In the polynomial division, we are only checking to see how many times x goes into x^2. Shouldn't we be checking how many times x + 3 goes into x^2 + 10x? After all, if this was arithmetic, we don't break up the divisor into pieces, like how we are doing with the polynomials. If this was arithmetic, if we couldn't divide it, then we add a decimal point and a zero until we could. How come in polynomials, we are allowed to break up the divisor in pieces like this?