Divide using synthetic division.
StartFraction x Superscript 5 Baseline plus 11 x cubed minus 12 Over x minus 1 EndFraction
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Part 1
StartFraction x Superscript 5 Baseline plus 11 x cubed minus 12 Over x minus 1 EndFraction
equals
  
enter your response here
​(Simplify your​ answer.)

To solve this problem using synthetic division, we first need to divide the polynomial \( x^5 + 11x^3 - 12 \) by \( x - 1 \). Here's how to perform synthetic division step by step.

### Step 1: Write the Coefficients
The given polynomial is \( x^5 + 0x^4 + 11x^3 + 0x^2 + 0x - 12 \), so the coefficients are:

\[
1 \quad 0 \quad 11 \quad 0 \quad 0 \quad -12
\]

The divisor is \( x - 1 \), which means the number to use for synthetic division is \( 1 \) (the root that would make \( x - 1 = 0 \)).

### Step 2: Set up the Synthetic Division
Write the number \( 1 \) outside and the coefficients inside:

\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 11 & 0 & 0 & -12 \\
  &   &   &   &   &   &    \\
\end{array}
\]

### Step 3: Perform Synthetic Division
1. Bring down the first coefficient \( 1 \).
2. Multiply \( 1 \times 1 \) and write it under the next coefficient.
3. Add the second coefficient and the product.

Repeat this process:

\[
\begin{array}{r|rrrrrr}
1 & 1 & 0 & 11 & 0 & 0 & -12 \\
  &   & 1 & 1 & 12 & 12 & 12 \\
\hline
  & 1 & 1 & 12 & 12 & 12 & 0 \\
\end{array}
\]

### Step 4: Write the Quotient
The quotient is formed by the new row of coefficients. So, the result of the division is:

\[
x^4 + x^3 + 12x^2 + 12x + 12
\]

Thus, the division of \( \frac{x^5 + 11x^3 - 12}{x - 1} \) simplifies to:

\[
x^4 + x^3 + 12x^2 + 12x + 12
\]

The remainder is \( 0 \), indicating that the division is exact.

---

Would you like further clarification on any part of this solution?

Here are five related questions to expand your understanding:
1. How do you handle synthetic division when the divisor is of the form \( x + a \)?
2. Can synthetic division be used for divisors with degrees higher than 1?
3. How does synthetic division work when there are missing terms in the dividend?
4. How would you perform synthetic division for a divisor of \( x - 2 \)?
5. What are the benefits of using synthetic division over long division for polynomials?

**Tip**: Always remember to include zeros for missing terms when setting up synthetic division, as it keeps the process organized and accurate.

Divide using synthetic division. StartFraction x Superscript 5 Baseline plus 11 x cubed minus 12 Over x minus 1 EndFraction. Simplify your answer.

Learn how to divide the polynomial x^5 + 11x^3 - 12 by x - 1 using synthetic division. Follow a detailed step-by-step solution, simplifying the expression to x^4 + x^3 + 12x^2 + 12x + 12. Suitable for high school students in Grades 9-12.

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