# more polynomials

• Nov 23rd 2009, 09:46 PM
ipokeyou
more polynomials

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 !
• Nov 24th 2009, 06:10 AM
Hello ipokeyou
Quote:

Originally Posted by ipokeyou

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 $x$ is divided by a quadratic in $x$ will be a linear expression in $x$: $ax+b$ (where either or both of $a$ and $b$ 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 $f(x)$ and that the quotient when it is divided by $(2x+1)(x-3)$ is $q(x)$. Then, since we know that the remainder is $3x-1$, we can write:
$f(x) = (2x+1)(x-3)q(x) + 3x -1$
which we can write as:
$f(x) = (2x+1)(x-3)q(x) + \frac32(2x +1) -\frac32-1$
$= (2x+1)(x-3)q(x) + \frac32(2x +1) -\frac52$
Now when this expression is divided by $(2x+1)$ the quotient will be the polynomial $(x-3)q(x) + \frac32$, and the remainder will be $-\frac52$. So there's your answer: $-\frac52=-2\tfrac12$.