https://file.aiquickdraw.com/mathbot/file/17266888041016x9dff5o.jpg

The given image shows a polynomial long division problem. The division expression is:

\[
\frac{z^2 - 19z - 11}{z + 8}
\]

Let’s solve this step by step using polynomial long division.

### Step 1: Divide the first term
Divide the first term of the numerator (\(z^2\)) by the first term of the denominator (\(z\)):

\[
\frac{z^2}{z} = z
\]

This gives the first term of the quotient, \(z\).

### Step 2: Multiply and subtract
Now, multiply \(z\) by the entire divisor \(z + 8\):

\[
z(z + 8) = z^2 + 8z
\]

Subtract this from the original polynomial:

\[
(z^2 - 19z - 11) - (z^2 + 8z) = -27z - 11
\]

### Step 3: Repeat the process
Now, divide the first term of the new expression \(-27z\) by \(z\):

\[
\frac{-27z}{z} = -27
\]

This gives the second term of the quotient, \(-27\).

### Step 4: Multiply and subtract again
Multiply \(-27\) by the divisor \(z + 8\):

\[
-27(z + 8) = -27z - 216
\]

Subtract this from the remaining expression:

\[
(-27z - 11) - (-27z - 216) = 205
\]

### Final Answer:
The quotient is \(z - 27\) and the remainder is 205, so the result of the division is:

\[
z - 27 + \frac{205}{z + 8}
\]

Would you like more details on any of these steps?

Here are 5 related questions for practice:
1. How do you perform synthetic division for a similar problem?
2. What is the relationship between polynomial division and factorization?
3. How can you verify the result of a polynomial division?
4. What happens if the divisor has a degree higher than 1?
5. How can you interpret the remainder in polynomial long division?

**Tip:** When subtracting polynomials, remember to distribute the negative sign carefully across all terms.

Perform the polynomial long division of (z^2 - 19z - 11) by (z + 8).

Solve the polynomial long division of z^2 - 19z - 11 by z + 8 with detailed steps. This problem covers key algebra concepts and polynomial division techniques, perfect for students in Grades 10-12.

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