<s>[INST] Synthetic division of polynomials by factor of the form $(ax+b)$ Find the quotient and remainder when $6x^4-11x^3+5x^2-7x+9$ is divided by $(2x-3)$. I expressed the divisor $(2x-3)$ as $2(x-\frac{3}{2})$ and conducted synthetic division by $\frac{3}{2}$ and obtained the coefficients of the quotients as $6, -2, 2$ and $-4$ and the remainder as $3$. Then I divided the quotients by $2$. Why should I do this? and why should I not divide the remainder by $2$?
 [/INST] In my opinion, you do not need to change $(2x-3)$ by anything. If you perform the long divition of $6x^4-11x^3+5x^2-7x+9$ by $(2x-3)$, you should directly obtain 
$$\frac{6x^4-11x^3+5x^2-7x+9 }{2x-3}=3 x^3-x^2+x-2+\frac{3}{2 x-3}$$ If i May suggest, don't make your life more complex than it is. Cheers.
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