use synthetic division to solve: (2x^4 - x^3 -21x^2 -26x -8)/(x^3 -x^2 -10x -8)

We are given the division:

\[
\frac{2x^4 - x^3 - 21x^2 - 26x - 8}{x^3 - x^2 - 10x - 8}
\]

We can use synthetic division to solve this. Synthetic division is often used to divide a polynomial by a linear binomial (like \(x - a\)). However, in this case, we are dividing by a cubic polynomial (\(x^3 - x^2 - 10x - 8\)), so synthetic division isn't the most appropriate method here. Instead, long division of polynomials would be the preferred approach.

Let's proceed with polynomial long division.

### Step-by-step Long Division:

1. **Divide the first term of the dividend by the first term of the divisor:**

   \[
   \frac{2x^4}{x^3} = 2x
   \]

2. **Multiply the entire divisor by 2x:**

   \[
   2x(x^3 - x^2 - 10x - 8) = 2x^4 - 2x^3 - 20x^2 - 16x
   \]

3. **Subtract the result from the original polynomial:**

   \[
   (2x^4 - x^3 - 21x^2 - 26x - 8) - (2x^4 - 2x^3 - 20x^2 - 16x)
   \]
   \[
   = (2x^4 - 2x^4) + (-x^3 + 2x^3) + (-21x^2 + 20x^2) + (-26x + 16x) - 8
   \]
   \[
   = x^3 - x^2 - 10x - 8
   \]

4. **Now repeat the process:**

   - Divide the new first term by the first term of the divisor:

     \[
     \frac{x^3}{x^3} = 1
     \]

   - Multiply the entire divisor by 1:

     \[
     1(x^3 - x^2 - 10x - 8) = x^3 - x^2 - 10x - 8
     \]

   - Subtract:

     \[
     (x^3 - x^2 - 10x - 8) - (x^3 - x^2 - 10x - 8) = 0
     \]

So, the quotient is \(2x + 1\) with a remainder of 0.

Thus, the result of the division is:

\[
\boxed{2x + 1}
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you perform synthetic division on a linear divisor?
2. What are the key differences between synthetic and polynomial long division?
3. Can synthetic division handle non-linear divisors?
4. How do you identify when to use synthetic division over long division?
5. What are some common errors to watch for during polynomial division?

### Tip:
When dividing polynomials, always ensure that all terms are in descending order of the degree and watch for any missing terms (use a zero coefficient if needed).

Learn how to solve the polynomial division of (2x^4 - x^3 - 21x^2 - 26x - 8) by (x^3 - x^2 - 10x - 8) using the long division method. This detailed step-by-step solution highlights key algebraic concepts for students in Grades 10-12.

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