Question

Given P(x) = 2x3 - 11x2 + 18x - 15 is divided by x – 3. Use synthetic division to find the quotient and the remainder.

Answer 1

You can find the remainder right away by simply plugging in $x=3$. The polynomial remainder theorem guarantees that the value of $p(3)$ is the remainder upon dividing $p(x)$ by $x-3$, but I digress...

Synthetic division yields

3   |   2   -11   18   -15
.    |           6   -15     9
- - - - - - - - - - - - - - - - -
.    |   2     -5     3     -6

which translates to

$\dfrac{2x^3-11x^2+18x-15}{x-3}=2x^2-5x+3-\dfrac6{x-3}$

(and note that $p(3)=2(3)^3-11(3)^2+18(3)-15=-6$, as expected)