https://file.aiquickdraw.com/mathbot/file/1728212481323dx799jyz.jpg

The image shows four polynomial division problems where the task is to use **remainder division** or **synthetic division** to find the remainder for each division. Let's go through them step by step.

### Problem 1:
\[ 4x^3 + 2x + 10 \div (x + 3) \]

**Using Synthetic Division**:

- The divisor is \(x + 3\), so we use \(-3\) for synthetic division.
- Coefficients of the dividend: \(4, 0, 2, 10\).

\[
\begin{array}{r|rrrr}
-3 & 4 & 0 & 2 & 10 \\
   &   & -12 & 36 & -114 \\
\hline
   & 4 & -12 & 38 & -104 \\
\end{array}
\]

**Remainder**: \(-104\).

### Problem 2:
\[ 2x^4 - 9x^3 + 14x^2 - 8 \div (x + 2) \]

**Using Synthetic Division**:

- The divisor is \(x + 2\), so we use \(-2\).
- Coefficients of the dividend: \(2, -9, 14, 0, -8\).

\[
\begin{array}{r|rrrrr}
-2 & 2 & -9 & 14 & 0 & -8 \\
   &   & -4 & 26 & -60 & 120 \\
\hline
   & 2 & -13 & 40 & -60 & 112 \\
\end{array}
\]

**Remainder**: \(112\).

### Problem 3:
\[ 6x^2 + 3x - 9 \div (x - 1) \]

**Using Synthetic Division**:

- The divisor is \(x - 1\), so we use \(1\).
- Coefficients of the dividend: \(6, 3, -9\).

\[
\begin{array}{r|rrr}
1 & 6 & 3 & -9 \\
  &   & 6 & 9 \\
\hline
  & 6 & 9 & 0 \\
\end{array}
\]

**Remainder**: \(0\) (this divides evenly).

### Problem 4:
\[ 2x^4 - 9x^3 + 14x^2 - 8 \div (x - 2) \]

**Using Synthetic Division**:

- The divisor is \(x - 2\), so we use \(2\).
- Coefficients of the dividend: \(2, -9, 14, 0, -8\).

\[
\begin{array}{r|rrrrr}
2 & 2 & -9 & 14 & 0 & -8 \\
  &   & 4 & -10 & 8 & 16 \\
\hline
  & 2 & -5 & 4 & 8 & 8 \\
\end{array}
\]

**Remainder**: \(8\).

---

Let me know if you'd like more detailed steps on any of the problems! Here are some follow-up questions you might find useful:

1. How does synthetic division differ from long division?
2. What happens if the remainder is zero in polynomial division?
3. Can synthetic division be used for divisors other than linear polynomials?
4. What is the relationship between the divisor and remainder in terms of factorization?
5. How can you check your work after performing synthetic division?

**Tip**: Synthetic division works best when dividing by linear divisors (like \(x - a\)). For higher-degree divisors, traditional long division is more appropriate.

Use remainder or synthetic division to find the remainder for the following polynomials: 1. 4x^3 + 2x + 10 ÷ (x + 3), 2. 2x^4 - 9x^3 + 14x^2 - 8 ÷ (x + 2), 3. 6x^2 + 3x - 9 ÷ (x - 1), 4. 2x^4 - 9x^3 + 14x^2 - 8 ÷ (x - 2)

Learn how to find the remainder using remainder or synthetic division for polynomials like 4x^3 + 2x + 10 ÷ (x + 3), 2x^4 - 9x^3 + 14x^2 - 8 ÷ (x + 2), and more. Step-by-step solutions provided. Suitable for high school algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation