(x⁵+x³+3x²+6)÷(x+2) synthetic division

To solve \(\frac{x^5 + x^3 + 3x^2 + 6}{x + 2}\) using synthetic division, follow these steps:

### Step 1: Set up the synthetic division
1. Write down the coefficients of the polynomial \(x^5 + x^3 + 3x^2 + 0x + 6\). Note that you need to include a coefficient for every power of \(x\) down to \(x^0\), even if the coefficient is zero:
   - Coefficients: \[1, 0, 1, 3, 0, 6\]

2. Write down the zero of the divisor \(x + 2\). To find the zero, set \(x + 2 = 0\) and solve for \(x\):
   - Zero: \(x = -2\)

### Step 2: Perform the synthetic division
- Bring down the first coefficient (1) as it is.
- Multiply it by the zero of the divisor (-2) and add the result to the next coefficient.

Here’s the step-by-step process:

\[
\begin{array}{r|rrrrrr}
-2 & 1 & 0 & 1 & 3 & 0 & 6 \\
   &   & -2 & 4 & -6 & 6 & -12 \\
\hline
   & 1 & -2 & 5 & -3 & 6 & -6 \\
\end{array}
\]

### Step 3: Interpret the result
The bottom row gives the coefficients of the quotient and the remainder:
- The quotient is \(1x^4 - 2x^3 + 5x^2 - 3x + 6\).
- The remainder is \(-6\).

### Final Answer
\[
\frac{x^5 + x^3 + 3x^2 + 6}{x + 2} = x^4 - 2x^3 + 5x^2 - 3x + 6 - \frac{6}{x + 2}
\]

Would you like more details on any of the steps? Here are five related questions you might consider:

1. How do you perform synthetic division with a divisor other than \(x + 2\)?
2. Can synthetic division be used for polynomials of any degree?
3. What happens if the remainder is zero in synthetic division?
4. How does synthetic division compare to long division for polynomials?
5. What is the significance of the remainder in synthetic division?

**Tip:** Remember that synthetic division only works when the divisor is a linear polynomial (i.e., of the form \(x - c\)).

The query submitted by the user: (x⁵+x³+3x²+6)÷(x+2) synthetic division

Learn how to solve the polynomial division of (x⁵ + x³ + 3x² + 6) by (x + 2) using synthetic division. This step-by-step solution covers key algebraic concepts, including synthetic division and the polynomial remainder theorem. Suitable for students in Grades 9-12.

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