Find the remainder when $x^3+ x^2 + x + 1$ is divided by $x - \frac{1}{2}$ using remainder theorem.

Given:

$x^3+ x^2 + x + 1$ is divided by $x - \frac{1}{2}$

To do:

Use remainder theorem to find the remainder when $x^3+ x^2 + x + 1$ is divided by $x - \frac{1}{2}$.

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 $p(a)$.

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

$g(x) = x -\frac{1}{2}$

So, the remainder will be $f(\frac{1}{2})$.

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

$= \frac{1}{8} + \frac{1}{4} + \frac{1}{2}+1$

$=\frac{1+1\times2+1\times4+1\times8}{8}$      (LCM of 1, 2, 4 and 8 is 8)

$=\frac{1+2+4+8}{8}$

$=\frac{15}{8}$

Therefore, the remainder is $\frac{15}{8}$.