What is the remainder when $(x^2-4x+7)$ is divided by $(x-2)$?

The correct option is 3.

To find the remainder when a polynomial is divided by a linear expression, we can use the Remainder Theorem.

The Remainder Theorem states that when a polynomial $P\left(x\right)$ is divided by $(x-c)$, the remainder is equal to $P\left(c\right)$.

In this problem, the polynomial is:
$P\left(x\right)=x^2-4x+7$

We are dividing by $(x-2)$. Comparing this to $(x-c)$, we identify that:
$c=2$

According to the theorem, the remainder is $P\left(2\right)$. We find this value by substituting $x=2$ into the polynomial:
$P\left(2\right)={\left(2\right)}^2-4\left(2\right)+7$

Now, calculate each term step-by-step:
First, calculate the square term:
$2^2=4$
Next, calculate the multiplication term:
$4\times 2=8$

Substitute these values back into the expression:
$P\left(2\right)=4-8+7$

Perform the addition and subtraction:
$4-8=-4$
$-4+7=3$

Therefore, the remainder when $(x^2-4x+7)$ is divided by $(x-2)$ is 3.