To solve the division of the polynomial [tex]\( 6x^2 - 13x + 6 \)[/tex] by [tex]\( 2x - 3 \)[/tex], we perform polynomial long division. Here's a detailed, step-by-step solution:
1. Setup the Division: Write the dividend and divisor.
[tex]\[
\frac{6x^2 - 13x + 6}{2x - 3}
\][/tex]
2. First Division Step: Divide the leading term of the numerator by the leading term of the denominator:
[tex]\[
\frac{6x^2}{2x} = 3x
\][/tex]
3. Multiply and Subtract: Multiply [tex]\( 3x \)[/tex] by [tex]\( 2x - 3 \)[/tex] and subtract the result from the original polynomial.
[tex]\[
(3x) \cdot (2x - 3) = 6x^2 - 9x
\][/tex]
[tex]\[
6x^2 - 13x + 6 - (6x^2 - 9x) = -13x + 9x + 6 = -4x + 6
\][/tex]
4. Second Division Step: Divide the leading term of the new polynomial by the leading term of the divisor:
[tex]\[
\frac{-4x}{2x} = -2
\][/tex]
5. Multiply and Subtract: Multiply [tex]\( -2 \)[/tex] by [tex]\( 2x - 3 \)[/tex] and subtract the result from the current polynomial.
[tex]\[
(-2) \cdot (2x - 3) = -4x + 6
\][/tex]
[tex]\[
-4x + 6 - (-4x + 6) = 0
\][/tex]
Since the remainder is [tex]\( 0 \)[/tex], the division is exact.
The quotient is:
[tex]\[
3x - 2
\][/tex]
So, the correct answer is:
[tex]\[
\boxed{B) \ 3x - 2}
\][/tex]
