https://file.aiquickdraw.com/mathbot/file/17261541521077x4p47sp.jpg

The given expression asks to find the difference between two rational expressions:

\[
\frac{x - 2}{x^2 + x - 6} - \frac{x - 3}{x^3 - 2x^2 - 9x + 18}
\]

### Step 1: Factor Denominators
We begin by factoring the denominators of both terms.

#### First Denominator: \( x^2 + x - 6 \)
We look for two numbers that multiply to \(-6\) and add to \(1\):
\[
x^2 + x - 6 = (x - 2)(x + 3)
\]

#### Second Denominator: \( x^3 - 2x^2 - 9x + 18 \)
This is a cubic expression, so we use factoring by grouping:
1. Factor out the greatest common factor from pairs of terms:
\[
x^3 - 2x^2 - 9x + 18 = x^2(x - 2) - 9(x - 2)
\]
2. Now, factor out \( (x - 2) \):
\[
(x^2 - 9)(x - 2)
\]
3. The expression \(x^2 - 9\) is a difference of squares:
\[
x^2 - 9 = (x - 3)(x + 3)
\]
Thus, the fully factored second denominator is:
\[
x^3 - 2x^2 - 9x + 18 = (x - 2)(x - 3)(x + 3)
\]

### Step 2: Rewrite the Rational Expressions
Now we rewrite the original expression with the factored denominators:
\[
\frac{x - 2}{(x - 2)(x + 3)} - \frac{x - 3}{(x - 2)(x - 3)(x + 3)}
\]

### Step 3: Simplify the First Fraction
In the first term, the factor \(x - 2\) appears in both the numerator and denominator, so we can cancel it:
\[
\frac{1}{x + 3} - \frac{x - 3}{(x - 2)(x - 3)(x + 3)}
\]

### Step 4: Find the Common Denominator
The common denominator between the two fractions is \((x - 2)(x - 3)(x + 3)\). We rewrite both fractions with this common denominator:
1. The first term becomes:
\[
\frac{1}{x + 3} = \frac{(x - 2)(x - 3)}{(x - 2)(x - 3)(x + 3)}
\]
2. The second term remains:
\[
\frac{x - 3}{(x - 2)(x - 3)(x + 3)}
\]

### Step 5: Subtract the Fractions
Now that we have a common denominator, we can subtract the numerators:
\[
\frac{(x - 2)(x - 3) - (x - 3)}{(x - 2)(x - 3)(x + 3)}
\]

### Step 6: Simplify the Numerator
Factor out \( (x - 3) \) from the numerator:
\[
(x - 3)\left((x - 2) - 1\right) = (x - 3)(x - 3)
\]
Thus, the numerator becomes:
\[
(x - 3)^2
\]

### Step 7: Final Expression
The resulting fraction is:
\[
\frac{(x - 3)^2}{(x - 2)(x - 3)(x + 3)}
\]
Cancel the common factor \(x - 3\):
\[
\frac{x - 3}{(x - 2)(x + 3)}
\]

### Final Answer:
\[
\frac{x - 3}{(x - 2)(x + 3)}
\]

This is the simplified expression after finding the difference.

Let me know if you'd like further clarification or have additional questions!

---

### Relative Questions:
1. How do you factor a cubic polynomial using grouping?
2. Why is it important to find a common denominator when subtracting rational expressions?
3. How do you simplify a rational expression by canceling common factors?
4. What is the process for factoring a difference of squares?
5. How do you determine when a factor can be canceled in a rational expression?

---

### Tip:
Always check for common factors in both the numerator and denominator after factoring to simplify your expressions effectively.

Find the difference between the two rational expressions: (x - 2) / (x^2 + x - 6) - (x - 3) / (x^3 - 2x^2 - 9x + 18).

Learn how to subtract rational expressions and factor both quadratic and cubic equations. This step-by-step solution covers factoring, finding common denominators, and simplifying rational expressions. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation