use a synthetic division table to solve $$ \frac{2x^3-8x^2-1}{x-4} $$

To solve the division of $ \frac{2x^3 - 8x^2 - 1}{x-4} $ using synthetic division, we follow these steps:

### Step-by-Step Solution Using Synthetic Division

1. **Set Up Synthetic Division:**
   - Write down the coefficients of the polynomial $2x^3 - 8x^2 + 0x - 1$ (note the zero coefficient for the $x$ term).
   - Use $4$ as the divisor since we are dividing by $x-4$.

2. **Synthetic Division Table:**

   \[
   \begin{array}{c|cccc}
   4 & 2 & -8 & 0 & -1 \\
   \hline
     &   & 8  & 0 & -1 \\
   \hline
     & 2 & 0 & 0 & -1 \\
   \end{array}
   \]

3. **Perform the Division:**
   - Bring down the first coefficient $2$ to the bottom row.
   - Multiply $2$ by $4$ (the divisor) and write the result ($8$) under the next coefficient $-8$.
   - Add the column: $-8 + 8 = 0$.
   - Repeat this process for the next coefficient.

Here's the step-by-step calculation:

\[
\begin{array}{r|rrrr}
4 & 2 & -8 & 0 & -1 \\
\hline
  &   & 8 & 32 & 128 \\
\hline
  & 2 & 0 & 32 & 127 \\
\end{array}
\]

4. **Interpret the Result:**
   - The bottom row (2, 0, 32, 127) represents the coefficients of the quotient polynomial and the remainder.
   - The quotient polynomial is $2x^2 + 0x + 32 = 2x^2 + 32$.
   - The remainder is $127$.

So, the result of the division is:

\[ 2x^2 + 32 + \frac{127}{x-4} \]

### Summary

The division of $ \frac{2x^3 - 8x^2 - 1}{x-4} $ results in:

\[ 2x^2 + 32 + \frac{127}{x-4} \]

Do you want more details or have any questions? Here are some related questions you might be interested in:

1. How is synthetic division different from long division in polynomials?
2. Can synthetic division be used for all types of polynomial division?
3. What are the advantages of using synthetic division?
4. How do you handle a remainder in polynomial division?
5. Can you apply synthetic division to polynomials with missing terms?
6. How do you interpret the results of synthetic division?
7. Can synthetic division be used for dividing by non-linear polynomials?
8. How does synthetic division relate to the Factor Theorem?

**Tip:** When setting up synthetic division, always ensure that all coefficients, including those of missing terms, are included to avoid errors in calculation.

Learn how to solve the division of 2x^3 - 8x^2 - 1 by (x-4) using synthetic division with a detailed step-by-step solution. This problem covers polynomial division concepts, specifically synthetic division. Suitable for advanced high school students (Grades 11-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation