Use Remainder theorem to find the remainder when $ f(x) $ is divided by $ g(x) $ in the following $f(x)=x^{2}-5 x+7, g(x)=x+3$.

Given:

$f(x)=x^2-5x+7$ and $g(x)=x+3$

To do:

Use remainder theorem to find the remainder when f(x) is divided by g(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 $p(a)$.

$f(x) = x^2 - 5x + 7$

$g (x) = x + 3$

$=x - (-3)$

So, the remainder will be $f(-3)$.

$f(-3) = (-3)^2 - 5(-3) + 7$

$= 9 + 15 + 7$

$= 31$

Therefore, the remainder is $31$.