To divide \(3x^2 + 10x - 8\) by \(3x - 2\), we can use polynomial long division.

### Step 1: Set up the division

We are dividing:
\[
\frac{3x^2 + 10x - 8}{3x - 2}
\]

### Step 2: Perform the division

1. **Divide the leading term** of the dividend \(3x^2\) by the leading term of the divisor \(3x\):
   \[
   \frac{3x^2}{3x} = x
   \]
   So the first term in the quotient is \(x\).

2. **Multiply** \(x\) by \(3x - 2\):
   \[
   x \cdot (3x - 2) = 3x^2 - 2x
   \]

3. **Subtract** this result from the original polynomial:
   \[
   (3x^2 + 10x - 8) - (3x^2 - 2x) = 12x - 8
   \]

4. **Divide** the leading term of the new polynomial \(12x\) by the leading term of the divisor \(3x\):
   \[
   \frac{12x}{3x} = 4
   \]
   So the next term in the quotient is \(+4\).

5. **Multiply** \(4\) by \(3x - 2\):
   \[
   4 \cdot (3x - 2) = 12x - 8
   \]

6. **Subtract** this result from the current polynomial:
   \[
   (12x - 8) - (12x - 8) = 0
   \]

### Final Answer:
The quotient is:
\[
x + 4
\]

Thus, \(\frac{3x^2 + 10x - 8}{3x - 2} = x + 4\).

Would you like further details or have any questions?

Here are five related questions:
1. How do you perform synthetic division for polynomials?
2. What is the remainder theorem and how does it apply here?
3. How can you factor polynomials to simplify division?
4. What is the relationship between polynomial division and finding roots?
5. Can polynomial long division be used for higher-degree polynomials?

**Tip:** Always double-check your subtraction steps in polynomial long division to avoid errors.

Learn how to divide the polynomial 3x^2 + 10x - 8 by 3x - 2 using polynomial long division. This step-by-step solution explains key algebraic concepts, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation