What is the quotient when 3x³+4²+8 is divided by x+2

To divide \( 3x^3 + 4x^2 + 8 \) by \( x + 2 \), we can use synthetic division or polynomial long division. Let's perform synthetic division:

### Step 1: Set up synthetic division
We divide \( 3x^3 + 4x^2 + 0x + 8 \) by \( x + 2 \). The divisor \( x + 2 \) means we use \( -2 \) for synthetic division. Write the coefficients of the polynomial:

- \( 3x^3 + 4x^2 + 0x + 8 \) gives coefficients \( 3, 4, 0, 8 \).

### Step 2: Perform synthetic division

\[
\begin{array}{r|rrrr}
-2 & 3 & 4 & 0 & 8 \\
   &   & -6 & 4 & -8 \\
\hline
   & 3 & -2 & 4 & 0 \\
\end{array}
\]

### Step 3: Interpret the result

The quotient is \( 3x^2 - 2x + 4 \), and the remainder is \( 0 \).

### Final Answer:
The quotient is \( 3x^2 - 2x + 4 \) with no remainder.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you perform polynomial long division?
2. What happens when there is a remainder in polynomial division?
3. Can synthetic division be used for divisors of higher degrees?
4. How is synthetic division different from regular division?
5. How can you factor the quotient further if needed?

**Tip:** When using synthetic division, always check for missing powers of \(x\) by adding zero as a placeholder.

What is the quotient when 3x³+4²+8 is divided by x+2?

Explore the detailed solution to dividing the polynomial 3x³ + 4x² + 8 by x + 2 using synthetic division. This solution covers key algebraic concepts, including synthetic division, and is suitable for students in Grades 10-12.

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