https://file.aiquickdraw.com/mathbot/file/172618294946432uq6o3f.jpg

To solve this synthetic division problem correctly, we need to divide the polynomial \( x^5 - 32 \) by \( x - 2 \) using synthetic division. Let's break this down step by step:

### Step 1: Set up the synthetic division
For the polynomial \( x^5 - 32 \), we can represent it as:
\[
x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + (-32)
\]
These coefficients are:
\[
[1, 0, 0, 0, 0, -32]
\]

We're dividing by \( x - 2 \), so the divisor is \( 2 \).

### Step 2: Perform the synthetic division

We will now perform synthetic division with \( 2 \) and the coefficients \([1, 0, 0, 0, 0, -32]\).

1. Bring down the first coefficient (1).
2. Multiply it by the divisor (2) and add it to the next coefficient.
3. Continue this process across all coefficients.

Let's compute it step-by-step.

\[
\begin{array}{r|rrrrrr}
2 & 1 & 0 & 0 & 0 & 0 & -32 \\
  &   & 2 & 4 & 8 & 16 & 32 \\
\hline
  & 1 & 2 & 4 & 8 & 16 & 0 \\
\end{array}
\]

### Step 3: Write the quotient and remainder
The bottom row shows the coefficients of the quotient:
\[
x^4 + 2x^3 + 4x^2 + 8x + 16
\]
The remainder is 0.

### Final Answer:
- **Quotient**: \( x^4 + 2x^3 + 4x^2 + 8x + 16 \)
- **Remainder**: 0

The original remainder in the photo (-16) seems to be incorrect, as the correct remainder is 0.

Would you like more details on any step, or clarification on the synthetic division process?

Here are some related questions to further explore:
1. How do you perform synthetic division with more complex polynomials?
2. What happens if the remainder isn't zero in synthetic division?
3. How does synthetic division relate to the factor theorem?
4. How do you identify factors of polynomials using synthetic division?
5. Can synthetic division be used with divisors of degree greater than 1?

**Tip**: Always remember to include missing terms with a coefficient of zero when setting up synthetic division!

Use synthetic division to find the quotient and remainder when x^5 - 32 is divided by x - 2.

Learn how to solve the synthetic division of x^5 - 32 by x - 2. This solution includes a step-by-step breakdown, showing how to find both the quotient and the remainder. Key concepts include synthetic division, the remainder theorem, and polynomial division. Suitable for Grades 10-12.

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