Polynomial division vs synthetic division

If I am simplifing an expression in which I cannot factor I can use polynomial division, I can also use synethic division to get the same answer. I know that I can only use synethic division when I am dividing by a linear equation.

When I change the leading coefficient on the linear equation I found that I have to divide my numbers when I do synthetic division by that number, why does this have to happen

example using poly division

(42x^2-33)/(7x+7) = 6x-6+9/7x+7

using synthetic I get 42x-42 with a remainder of 9

so I can say that r= 9/x+1 Is the reason that I divide by 7 because I take that out a 7 and use -1 so I have to put the 7 back in so I have to divide everything by 7 to get the same answer when I use poly division?

If I am using synthetic division to solve my answer will I always divide my two numbers by the leading cooefficient on my linear term.

like if I want to divide 2x^2-17x-38 by 2x+3 if I use synthetic I will just divide by 2 when I am done

50k^3+10k^2-35k-7 divided by 5k-4 when I do synthetic divide all terms by 5?