use synthetic division to find the result when (2x^{3}-3x^{2}-18x + 27) is divided by (x - 3).

Explanation:

Step1: Set up synthetic division

Write the zero of the divisor (x - 3) (which is (3)) and the coefficients of the polynomial (2x^{3}-3x^{2}-18x + 27) (coefficients: (2,-3,-18,27)).

Step2: Bring down the first coefficient

Bring down the first coefficient (2). $$ \begin{array}{r|rrrr} 3&2&-3&-18&27\ &&&\ \hline &2&&& \end{array} $$

Step3: Multiply and add

Multiply (3) by (2) to get (6), add to (-3): (-3+6 = 3). $$ \begin{array}{r|rrrr} 3&2&-3&-18&27\ &&6&&\ \hline &2&3&& \end{array} $$ Multiply (3) by (3) to get (9), add to (-18): (-18 + 9=-9). $$ \begin{array}{r|rrrr} 3&2&-3&-18&27\ &&6&9&\ \hline &2&3&-9& \end{array} $$ Multiply (3) by (-9) to get (-27), add to (27): (27+(-27)=0). $$ \begin{array}{r|rrrr} 3&2&-3&-18&27\ &&6&9&-27\ \hline &2&3&-9&0 \end{array} $$

The coefficients of the quotient polynomial are (2,3,-9) and the degree of the quotient is (3 - 1=2).

Answer:

The quotient is (2x^{2}+3x - 9) and the remainder is (0). So (2x^{3}-3x^{2}-18x + 27=(x - 3)(2x^{2}+3x - 9))