https://file.aiquickdraw.com/mathbot/file/1723046473928zq8t38dd.jpg

Let's tackle the problems in the provided image. We will simplify the rational expressions and state all restrictions.

### Problem 1(a)
Simplify the following rational expression and state all restrictions:

\[
\frac{2x^2 + 7x + 3}{x^2 - 4x + 4} \times \frac{4x^2 - 8x}{2x^3 + 3x + 1} + \frac{x^3 - 9x}{6x + 6}
\]

#### Step 1: Factorize each polynomial
1. \(2x^2 + 7x + 3\): 
\[ 
2x^2 + 7x + 3 = (2x + 1)(x + 3)
\]

2. \(x^2 - 4x + 4\):
\[
x^2 - 4x + 4 = (x - 2)^2
\]

3. \(4x^2 - 8x\):
\[
4x^2 - 8x = 4x(x - 2)
\]

4. \(2x^3 + 3x + 1\):
This does not factor nicely, so we leave it as is.

5. \(x^3 - 9x\):
\[
x^3 - 9x = x(x^2 - 9) = x(x + 3)(x - 3)
\]

6. \(6x + 6\):
\[
6x + 6 = 6(x + 1)
\]

#### Step 2: Substitute the factors
\[
\frac{(2x + 1)(x + 3)}{(x - 2)^2} \times \frac{4x(x - 2)}{2x^3 + 3x + 1} + \frac{x(x + 3)(x - 3)}{6(x + 1)}
\]

#### Step 3: Simplify the expressions
Cancel the common factors:
\[
\frac{(2x + 1)(x + 3) \cdot 4x(x - 2)}{(x - 2)^2(2x^3 + 3x + 1)} = \frac{4x(2x + 1)(x + 3)}{(x - 2)(2x^3 + 3x + 1)}
\]

For the second part:
\[
\frac{x(x + 3)(x - 3)}{6(x + 1)}
\]

#### Step 4: Combine the expressions
We need a common denominator to combine the two terms. The least common denominator (LCD) is \((x - 2)(2x^3 + 3x + 1)\):

\[
\frac{4x(2x + 1)(x + 3) + x(x + 3)(x - 3)}{(x - 2)(2x^3 + 3x + 1)}
\]

Combine and simplify the numerator if possible.

#### Restrictions
The restrictions are values that make the denominators zero:
1. \(x^2 - 4x + 4 = 0 \Rightarrow (x - 2)^2 = 0 \Rightarrow x = 2\)
2. \(2x^3 + 3x + 1 = 0 \) (solve for x)
3. \(6x + 6 = 0 \Rightarrow x = -1\)

### Problem 1(b)
Simplify the following rational expression and state all restrictions:

\[
\frac{\frac{x-3}{x^2 - x - 2}}{\frac{x-4}{x^2 - x - 6}} + 1
\]

#### Step 1: Factorize each polynomial
1. \(x^2 - x - 2\):
\[
x^2 - x - 2 = (x - 2)(x + 1)
\]

2. \(x^2 - x - 6\):
\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

#### Step 2: Substitute the factors
\[
\frac{\frac{x-3}{(x - 2)(x + 1)}}{\frac{x-4}{(x - 3)(x + 2)}} + 1
\]

#### Step 3: Simplify the complex fraction
\[
\frac{(x-3)(x-3)(x+2)}{(x-4)(x-2)(x+1)} + 1
\]

#### Step 4: Combine and simplify
Simplify the numerator and denominator separately and then combine:

\[
\frac{(x-3)^2(x+2)}{(x-4)(x-2)(x+1)} + 1
\]

Combine with the 1:
\[
= \frac{(x-3)^2(x+2) + (x-4)(x-2)(x+1)}{(x-4)(x-2)(x+1)}
\]

#### Restrictions
The restrictions are values that make the denominators zero:
1. \(x^2 - x - 2 = 0 \Rightarrow (x - 2)(x + 1) = 0 \Rightarrow x = 2, -1\)
2. \(x^2 - x - 6 = 0 \Rightarrow (x - 3)(x + 2) = 0 \Rightarrow x = 3, -2\)

### Summary of restrictions:
- For 1(a): \(x = 2, -1\) and the roots of \(2x^3 + 3x + 1 = 0\)
- For 1(b): \(x = 2, -1, 3, -2\)

Would you like a detailed step-by-step breakdown for any specific part of the solution or any additional explanations? 

Here are 8 related questions for further exploration:

1. How do you determine the least common denominator (LCD) in rational expressions?
2. What are the methods to factorize cubic polynomials?
3. How do you simplify complex fractions?
4. What are the steps to combine rational expressions?
5. How do you find the restrictions in a rational expression?
6. How do you solve higher-degree polynomial equations?
7. What is the significance of restrictions in rational expressions?
8. How can you verify the solution of a simplified rational expression?

**Tip:** Always double-check for common factors in both the numerator and the denominator to simplify the rational expressions effectively.

Learn how to simplify complex rational expressions with detailed step-by-step solutions. This problem involves factoring polynomials, finding least common denominators, and identifying restrictions. Suitable for advanced high school 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