Hello ipokeyou Originally Posted by
ipokeyou Hi can someone please help me figure this out?
WHen a polynomial is divided by (2x + 1)(x - 3), the remainder is 3x - 1. What is the remainder when the polynomial is divided by 2x +1?
Thanks !
The remainder when any polynomial in $\displaystyle x$ is divided by a quadratic in $\displaystyle x$ will be a linear expression in $\displaystyle x$: $\displaystyle ax+b$ (where either or both of $\displaystyle a$ and $\displaystyle b$ may be zero); in other words the remainder will be of degree at most 1.
When it is divided by a linear expression, the remainder will be a constant term (which, again, may be zero).
So let's suppose that the polynomial is $\displaystyle f(x)$ and that the quotient when it is divided by $\displaystyle (2x+1)(x-3)$ is $\displaystyle q(x)$. Then, since we know that the remainder is $\displaystyle 3x-1$, we can write:$\displaystyle f(x) = (2x+1)(x-3)q(x) + 3x -1$
which we can write as:$\displaystyle f(x) = (2x+1)(x-3)q(x) + \frac32(2x +1) -\frac32-1$$\displaystyle = (2x+1)(x-3)q(x) + \frac32(2x +1) -\frac52$
Now when this expression is divided by $\displaystyle (2x+1)$ the quotient will be the polynomial $\displaystyle (x-3)q(x) + \frac32$, and the remainder will be $\displaystyle -\frac52$. So there's your answer: $\displaystyle -\frac52=-2\tfrac12$.
Grandad