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Let's solve the division of \( x^3 - 90x - 95 \) by \( x + 9 \) using synthetic division.

### Step-by-Step Solution:

1. **Set up the synthetic division**:
   - The divisor is \( x + 9 \), so we take the opposite of the coefficient of \( x \), which is \( -9 \).
   - The coefficients of the polynomial \( x^3 - 90x - 95 \) are \( [1, 0, -90, -95] \) because there is no \( x^2 \)-term (hence coefficient of 0).

2. **Synthetic division process**:
   - Write \( -9 \) on the left side, and the coefficients \( 1, 0, -90, -95 \) across the top.

\[
\begin{array}{r|rrrr}
-9 & 1 & 0 & -90 & -95 \\
   &   & -9 & 81 & 81 \\
\hline
   & 1 & -9 & -9 & -14 \\
\end{array}
\]

3. **Interpret the result**:
   - The quotient \( q(x) \) is \( 1x^2 - 9x - 9 \), and the remainder \( r(x) \) is \( -14 \).
   
So, the division can be written as:
\[
x^3 - 90x - 95 = (x + 9)(x^2 - 9x - 9) - 14
\]

Would you like more details on any step? 

### Related Questions:
1. What is synthetic division, and when is it typically used?
2. How can the remainder theorem be used to verify the remainder in synthetic division?
3. What are the steps to check the quotient and remainder by multiplying the divisor and quotient and adding the remainder?
4. How do we interpret the remainder in polynomial division?
5. How can synthetic division be adapted for dividing by other polynomials like \( x - c \)?

### Tip:
Always remember to write any missing terms with a zero coefficient during synthetic division, like we did for \( x^2 \) in this example.

Divide x^3 - 90x - 95 by x + 9 using synthetic division. Write the polynomial in the form: d(x)q(x) + r(x) (in other words as (x + 9) × q(x) + r(x)).

Learn how to divide the polynomial x^3 - 90x - 95 by x + 9 using synthetic division. This step-by-step solution breaks down the process into manageable steps and explains how to express the result in quotient and remainder form. Suitable for high school students studying polynomial division.

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