Find the remainder when $x^3+ 3x^2 + 3x + 1$ is divided by $ 5+2 x $

Given:

$x^3+ 3x^2 + 3x + 1$ is divided by $5+2 x$

To do:

We have to find the remainder when $x^3+ 3x^2 + 3x + 1$ is divided by $5+2 x$.

Solution:

The remainder theorem states that when a polynomial, $p(x)$ is divided by a linear polynomial, $x - a$ the remainder of that division will be equivalent to $p(a)$.

Let $f(x) = x^3+ 3x^2 + 3x + 1$

$g(x)= 5+2 x$

$2x+5=0$

$2x=-5$

$x=\frac{-5}{2}$

So, the remainder will be $f(\frac{-5}{2})$.

$f(\frac{-5}{2}) = (\frac{-5}{2})^3+3(\frac{-5}{2})^2+3(\frac{-5}{2}) + 1$

$= \frac{-125}{8}+3(\frac{25}{4})-\frac{15}{2}+1$

$=\frac{-125+75(2)-15(4)+1(8)}{8}$

$=\frac{-125+150-60+8}{8}$

$=\frac{-27}{8}$

Therefore, the remainder is $\frac{-27}{8}$.