Use the long division method to find the result when 6, x, cubed, minus, 7, x, squared, plus, 23, x, minus, 76x 
3
 −7x 
2
 +23x−7 is divided by 3, x, minus, 23x−2. If there is a remainder, express the result in the form q, left parenthesis, x, right parenthesis, plus, start fraction, r, left parenthesis, x, right parenthesis, divided by, b, left parenthesis, x, right parenthesis, end fractionq(x)+ 
r(x)/b(x)
​
 .

To divide the polynomial \( 6x^3 - 7x^2 + 23x - 76 \) by \( 3x - 2 \) using long division, we follow these steps:

1. **Setup the Division**: Write the dividend \( 6x^3 - 7x^2 + 23x - 76 \) and the divisor \( 3x - 2 \).

2. **Divide the Leading Terms**: Divide the leading term of the dividend \( 6x^3 \) by the leading term of the divisor \( 3x \):
   \[
   \frac{6x^3}{3x} = 2x^2
   \]
   This is the first term of the quotient.

3. **Multiply and Subtract**: Multiply the entire divisor \( 3x - 2 \) by \( 2x^2 \):
   \[
   2x^2(3x - 2) = 6x^3 - 4x^2
   \]
   Now, subtract this from the original polynomial:
   \[
   (6x^3 - 7x^2) - (6x^3 - 4x^2) = -3x^2
   \]
   Bring down the next term, \( +23x \):
   \[
   -3x^2 + 23x = -3x^2 + 23x
   \]

4. **Repeat the Process**: Divide the new leading term \( -3x^2 \) by \( 3x \):
   \[
   \frac{-3x^2}{3x} = -x
   \]
   Multiply and subtract:
   \[
   -x(3x - 2) = -3x^2 + 2x
   \]
   Subtract:
   \[
   (-3x^2 + 23x) - (-3x^2 + 2x) = 21x
   \]
   Bring down the next term, \( -76 \):
   \[
   21x - 76
   \]

5. **Final Step**: Divide \( 21x \) by \( 3x \):
   \[
   \frac{21x}{3x} = 7
   \]
   Multiply and subtract:
   \[
   7(3x - 2) = 21x - 14
   \]
   Subtract:
   \[
   (21x - 76) - (21x - 14) = -62
   \]

6. **Conclusion**: The division yields:
   \[
   q(x) = 2x^2 - x + 7
   \]
   with a remainder:
   \[
   r(x) = -62
   \]

The result can be expressed as:
\[
q(x) + \frac{r(x)}{b(x)} = 2x^2 - x + 7 - \frac{62}{3x - 2}
\]

### Final Answer:
\[
2x^2 - x + 7 - \frac{62}{3x - 2}
\]

Would you like more details or have any questions about this process? Here are some related questions you might consider:

1. What are the steps involved in polynomial long division?
2. How can you check the result of polynomial division?
3. What does the remainder represent in polynomial division?
4. How does this method apply to different degrees of polynomials?
5. Can polynomial long division be used for complex polynomials as well? 

**Tip**: Always keep track of your signs during subtraction, as mistakes here are common.

Use the long division method to find the result when 6x^3 - 7x^2 + 23x - 76 is divided by 3x - 2. If there is a remainder, express the result in the form q(x) + r(x)/b(x).

Explore the long division of polynomials with the example of dividing 6x^3 - 7x^2 + 23x - 76 by 3x - 2. This detailed guide covers each step of the division process and provides insights into the remainder. Ideal for high school students (Grades 9-12).

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