Let's solve number 2, which involves showing that \( 3x - 2 \) is a factor of \( 3x^3 - 8x^2 + 5x - 2 \) and then solving the cubic equation \( 3x^3 - 8x^2 + 5x - 2 = 0 \) for different types of numbers.

### **Step 1: Show that \( 3x - 2 \) is a Factor of \( 3x^3 - 8x^2 + 5x - 2 \)**

To determine if \( 3x - 2 \) is a factor, we can use polynomial division or synthetic division.

#### **Method 1: Polynomial Division**

We'll divide \( 3x^3 - 8x^2 + 5x - 2 \) by \( 3x - 2 \):

1. **First Term:**
   \[
   \frac{3x^3}{3x} = x^2
   \]
   Multiply \( x^2 \) by \( 3x - 2 \):
   \[
   x^2(3x - 2) = 3x^3 - 2x^2
   \]
   Subtract:
   \[
   (3x^3 - 8x^2 + 5x - 2) - (3x^3 - 2x^2) = -6x^2 + 5x - 2
   \]

2. **Second Term:**
   \[
   \frac{-6x^2}{3x} = -2x
   \]
   Multiply \( -2x \) by \( 3x - 2 \):
   \[
   -2x(3x - 2) = -6x^2 + 4x
   \]
   Subtract:
   \[
   (-6x^2 + 5x - 2) - (-6x^2 + 4x) = x - 2
   \]

3. **Third Term:**
   \[
   \frac{x}{3x} = \frac{1}{3}
   \]
   Multiply \( \frac{1}{3} \) by \( 3x - 2 \):
   \[
   \frac{1}{3}(3x - 2) = x - \frac{2}{3}
   \]
   Subtract:
   \[
   (x - 2) - (x - \frac{2}{3}) = -\frac{4}{3}
   \]

Since the remainder is not zero, there is an error in the manual calculation or interpretation of the polynomial division method; therefore, let me use another reliable method, the **Factor Theorem**.

#### **Method 2: Factor Theorem**

The **Factor Theorem** states that \( x - a \) is a factor of a polynomial \( P(x) \) if \( P(a) = 0 \).

Let's substitute \( x = \frac{2}{3} \) (since \( 3x - 2 = 0 \) when \( x = \frac{2}{3} \)) into the polynomial \( P(x) = 3x^3 - 8x^2 + 5x - 2 \):

\[
P\left(\frac{2}{3}\right) = 3\left(\frac{2}{3}\right)^3 - 8\left(\frac{2}{3}\right)^2

Explore how to solve the cubic equation 3x^3 - 8x^2 + 5x - 2 using factor theorem and polynomial division methods. This problem involves advanced concepts in algebra and cubic equations, suitable for advanced high school students and beyond.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation