To find the quotient of [tex]\((2x^3 - 3x^2 - 14x + 15) \div (x - 3)\)[/tex], we perform polynomial long division. Let's go through the process step-by-step.
1. Set up the division:
[tex]\[
\frac{2x^3 - 3x^2 - 14x + 15}{x - 3}
\][/tex]
2. Divide the first term of the numerator by the first term of the divisor:
[tex]\[
\frac{2x^3}{x} = 2x^2
\][/tex]
3. Multiply the divisor by the result from step 2 and subtract from the numerator:
[tex]\[
(2x^3 - 3x^2 - 14x + 15) - (2x^2 \cdot (x - 3)) = 2x^3 - 3x^2 - 14x + 15 - (2x^3 - 6x^2) = 3x^2 - 14x + 15
\][/tex]
4. Repeat the previous steps for the new polynomial (3x^2 - 14x + 15):
[tex]\[
\frac{3x^2}{x} = 3x
\][/tex]
[tex]\[
(3x^2 - 14x + 15) - (3x \cdot (x - 3)) = 3x^2 - 14x + 15 - (3x^2 - 9x) = -5x + 15
\][/tex]
5. Repeat again for the resulting polynomial (-5x + 15):
[tex]\[
\frac{-5x}{x} = -5
\][/tex]
[tex]\[
(-5x + 15) - (-5 \cdot (x - 3)) = -5x + 15 - (-5x + 15) = 0
\][/tex]
Since the remainder is 0, we have completed the division, and the quotient is:
[tex]\[
2x^2 + 3x - 5
\][/tex]
Therefore, the correct answer is:
[tex]\[
2x^2 + 3x - 5
\][/tex]
So, the quotient [tex]\((2x^3 - 3x^2 - 14x + 15) \div (x - 3)\)[/tex] is:
[tex]\[
\boxed{2x^2 + 3x - 5}
\][/tex]
