To find the quotient and remainder when dividing [tex]\( x^2 - 4x + 8 \)[/tex] by [tex]\( x - 3 \)[/tex], we will use polynomial long division.
1. Set up the division:
[tex]\[
\frac{x^2 - 4x + 8}{x - 3}
\][/tex]
2. Divide the first term of the numerator by the first term of the denominator:
[tex]\[
\frac{x^2}{x} = x
\][/tex]
3. Multiply the whole divisor by this result (x) and subtract from the original numerator:
[tex]\[
(x^2 - 4x + 8) - (x \cdot (x - 3)) = (x^2 - 4x + 8) - (x^2 - 3x) = (-4x + 8) - (-3x) = -4x + 3x + 8 = -x + 8
\][/tex]
4. Repeat the process with the new polynomial (-x + 8):
[tex]\[
\frac{-x}{x} = -1
\][/tex]
5. Multiply the whole divisor by this result (-1) and subtract from the current polynomial:
[tex]\[
(-x + 8) - (-1 \cdot (x - 3)) = (-x + 8) - (-x + 3) = 8 - 3 = 5
\][/tex]
After these steps, the quotient is:
[tex]\[
\boxed{x - 1}
\][/tex]
and the remainder is:
[tex]\[
\boxed{5}
\][/tex]
