Divide 12 x cube minus x square - 13 x minus 6 by x minus 2 by long division

To divide the polynomial \(12x^3 - x^2 - 13x - 6\) by \(x - 2\) using long division, follow these steps:

1. **Set up the division**:
   
   \[
   \begin{array}{r|rr}
      & x - 2 & \quad 12x^3 - x^2 - 13x - 6 \\
   \end{array}
   \]

2. **Divide the first term** of the dividend by the first term of the divisor:
   
   \[
   \frac{12x^3}{x} = 12x^2
   \]
   
   Write \(12x^2\) above the division line.

3. **Multiply** \(12x^2\) by \(x - 2\):
   
   \[
   12x^2 \cdot (x - 2) = 12x^3 - 24x^2
   \]
   
   Write \(12x^3 - 24x^2\) under the dividend and subtract:

   \[
   \begin{array}{r|rr}
      & x - 2 & \quad 12x^3 - x^2 - 13x - 6 \\
      \hline
      & & \quad 12x^2 \\
      & & \quad 12x^3 - 24x^2 \\
      \hline
      & & \quad 23x^2 - 13x - 6 \\
   \end{array}
   \]

4. **Repeat the process** with the new polynomial \(23x^2 - 13x - 6\):

   - Divide the first term by \(x\):
     
     \[
     \frac{23x^2}{x} = 23x
     \]
     
     Write \(23x\) above the division line.

   - Multiply \(23x\) by \(x - 2\):
     
     \[
     23x \cdot (x - 2) = 23x^2 - 46x
     \]
     
     Write \(23x^2 - 46x\) under the new polynomial and subtract:

     \[
     \begin{array}{r|rr}
        & x - 2 & \quad 23x^2 - 13x - 6 \\
        \hline
        & & \quad 23x \\
        & & \quad 23x^2 - 46x \\
        \hline
        & & \quad 33x - 6 \\
     \end{array}
     \]

5. **Repeat the process** with the new polynomial \(33x - 6\):

   - Divide the first term by \(x\):
     
     \[
     \frac{33x}{x} = 33
     \]
     
     Write \(33\) above the division line.

   - Multiply \(33\) by \(x - 2\):
     
     \[
     33 \cdot (x - 2) = 33x - 66
     \]
     
     Write \(33x - 66\) under the new polynomial and subtract:

     \[
     \begin{array}{r|rr}
        & x - 2 & \quad 33x - 6 \\
        \hline
        & & \quad 33 \\
        & & \quad 33x - 66 \\
        \hline
        & & \quad 60 \\
     \end{array}
     \]

6. The remainder is \(60\).

Thus, the result of dividing \(12x^3 - x^2 - 13x - 6\) by \(x - 2\) is:

\[
12x^2 + 23x + 33 + \frac{60}{x-2}
\]

Would you like more details or have any questions?

Here are some related questions:

1. How do you perform synthetic division on the same polynomial?
2. What are the steps to check if a division result is correct?
3. How do you factor the given polynomial \(12x^3 - x^2 - 13x - 6\)?
4. Can you perform polynomial long division for \(x^4 + 3x^3 - 5x + 7\) divided by \(x + 2\)?
5. How do you interpret the remainder when dividing polynomials?
6. What are some applications of polynomial division in real-life problems?
7. How does polynomial division relate to the factor and remainder theorems?
8. What are the similarities and differences between polynomial long division and numerical long division?

**Tip:** Always double-check each step in polynomial long division to ensure accuracy, especially when subtracting terms.

Learn how to divide the polynomial 12x^3 - x^2 - 13x - 6 by x - 2 using long division. Follow a detailed step-by-step guide with examples and explanations. Suitable for high school students studying polynomial division.

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