Intuition for polynomial long division So I am trying to learn some basic algebra and go over some precalc material and I am terribly confused on why or how polynomial long division works.
I have looked at many sources online and they seem to all suggest that division of numbers is identical to division of polynomials since a number in its decimal expansion can be viewed as a sort of polynomial e.g $a_n10^n+\cdots+a_2100+10a_1+a_0$
What i don't understand is why we only use the term with the highest degree in the denominator as the divisor for the whole polynomial as shown here  Polynomial Long Division whereas with normal division we divide the whole divisor into the dividend for example when calculating $88÷32$ we look at how many times $32$ goes into $88$ and not how many times $30$ does whilst with polynomial division only the highest degree term is considered which corresponds to 30 in this example ignoring 2.
So how are they the same? Some intuition behind why polynomial division works would be greatly appreciated.
Thanks in advance.

Hint: in the algebra of polynomials there is no concept of "carrying" or "borrowing" when we add two polynomials, because there is no constraint on the size of the coefficients. In the algebra of integers expressed as decimals, when we add or subtract two decimal representations, we have to make adjustments to make the digits (coefficents) lie between $0$ and $9$. The intuition behind polynomial division is the same as the intuition behind integer division (subtract the biggest allowable multiple on each iteration), but the implementation is simpler - because there is no trial and error required to find the the biggest allowable multiple.
