Question

Can $(x-1)$ be the remainder on division of a polynomial $p\left(x\right)$ by $2x+3$?

• A. Yes
• B. No
• C. Cannot be justified
• D. None of above

Answer 1

Answer: (B)

No

Division algorithm stated that a polynomial $f\left(x\right)$ can written as
$f\left(x\right)=g\left(x\right)q+r$ where $q$ and $r$ are unique integer and $0<=r<g\left(x\right)$.
Here,
$g\left(x\right)=2x+3$ and $r\left(x\right)=x-1$
The power of the remainder is always less than the power of the divisor
Here, the degree of remainder is $1$ and the degree of divisor is $1$, which is not possible. Thus, $(x-1)$ cannot be the remainder of $p\left(x\right)$ when divided by $(2x+3)$