I really dont understand polynomial long division?
When you do regular long division (with numbers) each digit represents a power of 10. Like, let’s say you had 5024 ÷ 32. Really this means (5 x 10³)+(0 x 10²)+(2 x 10¹)+(4 x 10⁰) divided by (3 x 10¹)+(2 x 10⁰). Do you see how the digits are really coefficients and notice that the (0 x 10²) digit MUST be in there to fill the gap (5024 isn’t the same as 524)? You could still perform the long division with the numbers written out in this long form, rather than as a compact number.
Long division with polynomials works EXACTLY the same way. Instead of powers of 10, you have powers of x. So, in this case you might have 5x³+0x²+2x¹+4x⁰ divided by 3x¹+2x⁰. Again, notice the 0 term MUST be included to fill the gap. You actually could just write this as 5024÷32 and do the division, then after finding the answer replace the powers of 10 with x, and your answer would be correct. That is, if you did the long division, 5024÷32=157. So, 5x³+0x²+2x¹+4x⁰ divided by 3x¹+2x⁰ equals 1x²+5x¹+7x⁰, which simplifies to x²+5x+7.
When doing numerical long division, you are doing EXACTLY the same thing as polynomial long division, you just don’t realize it, because it looks so much neater and compact.