https://r2.erweima.ai/mathbot/image/1725958187812sb5swa6y.jpg

The image contains two sets of problems related to polynomial division. The first set requires **long division**, and the second set requires **synthetic division**.

### Long Division Problems:
1. \(\frac{2x^3 + 9x^2 + 3x - 4}{x + 4}\)
2. \(\frac{x^3 - 5x^2 + 2x - 10}{x^2 + 2}\)

### Synthetic Division Problems:
3. \(\frac{4x^2 + x - 3}{x - 3}\)
4. \(\frac{x^4 - 2x^3 + 3x^2 + 4x - 4}{x + 2}\)

Let's solve each problem step by step.

### 1. Long Division: \(\frac{2x^3 + 9x^2 + 3x - 4}{x + 4}\)

We perform polynomial long division:

- Divide the first term of the dividend \(2x^3\) by the first term of the divisor \(x\), which gives \(2x^2\).
- Multiply \(2x^2\) by \(x + 4\) to get \(2x^3 + 8x^2\), and subtract it from the original polynomial.
  \[
  (2x^3 + 9x^2 + 3x - 4) - (2x^3 + 8x^2) = x^2 + 3x - 4
  \]
- Divide \(x^2\) by \(x\) to get \(x\), then multiply \(x\) by \(x + 4\) to get \(x^2 + 4x\), and subtract:
  \[
  (x^2 + 3x - 4) - (x^2 + 4x) = -x - 4
  \]
- Divide \(-x\) by \(x\) to get \(-1\), multiply by \(x + 4\) to get \(-x - 4\), and subtract:
  \[
  (-x - 4) - (-x - 4) = 0
  \]

Thus, the quotient is:
\[
\boxed{2x^2 + x - 1}
\]

### 2. Long Division: \(\frac{x^3 - 5x^2 + 2x - 10}{x^2 + 2}\)

We perform polynomial long division:

- Divide the first term \(x^3\) by the first term \(x^2\), which gives \(x\).
- Multiply \(x\) by \(x^2 + 2\) to get \(x^3 + 2x\), and subtract:
  \[
  (x^3 - 5x^2 + 2x - 10) - (x^3 + 2x) = -5x^2 - 10
  \]
- Divide \(-5x^2\) by \(x^2\), which gives \(-5\).
- Multiply \(-5\) by \(x^2 + 2\) to get \(-5x^2 - 10\), and subtract:
  \[
  (-5x^2 - 10) - (-5x^2 - 10) = 0
  \]

Thus, the quotient is:
\[
\boxed{x - 5}
\]

### 3. Synthetic Division: \(\frac{4x^2 + x - 3}{x - 3}\)

We perform synthetic division using the root \(x = 3\):

- The coefficients are \(4, 1, -3\).
- Bring down the first coefficient \(4\).
- Multiply \(4\) by \(3\) to get \(12\), add to \(1\) to get \(13\).
- Multiply \(13\) by \(3\) to get \(39\), add to \(-3\) to get \(36\).

Thus, the quotient is:
\[
\boxed{4x + 13} \text{ with a remainder of } 36
\]

### 4. Synthetic Division: \(\frac{x^4 - 2x^3 + 3x^2 + 4x - 4}{x + 2}\)

We perform synthetic division using the root \(x = -2\):

- The coefficients are \(1, -2, 3, 4, -4\).
- Bring down the first coefficient \(1\).
- Multiply \(1\) by \(-2\) to get \(-2\), add to \(-2\) to get \(-4\).
- Multiply \(-4\) by \(-2\) to get \(8\), add to \(3\) to get \(11\).
- Multiply \(11\) by \(-2\) to get \(-22\), add to \(4\) to get \(-18\).
- Multiply \(-18\) by \(-2\) to get \(36\), add to \(-4\) to get \(32\).

Thus, the quotient is:
\[
\boxed{x^3 - 4x^2 + 11x - 18} \text{ with a remainder of } 32
\]

---

Let me know if you want any further details or explanations! Here are five related questions you could explore:

1. How does synthetic division differ from long division in terms of steps and usability?
2. What are some applications of polynomial division in calculus?
3. How can you verify the remainder when dividing polynomials?
4. Can synthetic division be used for divisors other than linear ones?
5. How would you perform polynomial division with complex coefficients?

**Tip**: Always check your division by multiplying the quotient with the divisor and adding the remainder to verify if you get the original polynomial.

Learn how to solve polynomial division problems using both long division and synthetic division. The problems include dividing cubic and quadratic polynomials with step-by-step solutions. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation