Given the functions:
f(x)=8x
g(x)=x+7
h(x)=3x^2 +23x+14
Determine each of the following.

Give your answers as simplified expressions written in descending order.
 

 
Find and simplify 
 g(x)+h(x)

 
Find and simplify 
 h(x)-g(x)

 
Find and simplify 
 f(x)times h(x)                                                                                                                                        
   Find and simplify h(x) / g(x)
, hint: you will need to factor h(x)	
 
 

 
The domain restriction for g(x) / f(x)
 
 is	
x cannot equal 

Let's solve each of the given expressions step by step.

### 1. Find and simplify \( g(x) + h(x) \)

We are given:
- \( g(x) = x + 7 \)
- \( h(x) = 3x^2 + 23x + 14 \)

To find \( g(x) + h(x) \), simply add these two functions:

\[
g(x) + h(x) = (x + 7) + (3x^2 + 23x + 14)
\]

Combine like terms:

\[
g(x) + h(x) = 3x^2 + (23x + x) + (14 + 7)
\]
\[
g(x) + h(x) = 3x^2 + 24x + 21
\]

So, the simplified expression is:

\[
g(x) + h(x) = 3x^2 + 24x + 21
\]

---

### 2. Find and simplify \( h(x) - g(x) \)

We are given:
- \( g(x) = x + 7 \)
- \( h(x) = 3x^2 + 23x + 14 \)

To find \( h(x) - g(x) \), subtract the two functions:

\[
h(x) - g(x) = (3x^2 + 23x + 14) - (x + 7)
\]

Distribute the negative sign and combine like terms:

\[
h(x) - g(x) = 3x^2 + 23x + 14 - x - 7
\]
\[
h(x) - g(x) = 3x^2 + (23x - x) + (14 - 7)
\]
\[
h(x) - g(x) = 3x^2 + 22x + 7
\]

So, the simplified expression is:

\[
h(x) - g(x) = 3x^2 + 22x + 7
\]

---

### 3. Find and simplify \( f(x) \times h(x) \)

We are given:
- \( f(x) = 8x \)
- \( h(x) = 3x^2 + 23x + 14 \)

To find \( f(x) \times h(x) \), multiply these two functions:

\[
f(x) \times h(x) = 8x(3x^2 + 23x + 14)
\]

Distribute \( 8x \) to each term in \( h(x) \):

\[
f(x) \times h(x) = 8x \times 3x^2 + 8x \times 23x + 8x \times 14
\]
\[
f(x) \times h(x) = 24x^3 + 184x^2 + 112x
\]

So, the simplified expression is:

\[
f(x) \times h(x) = 24x^3 + 184x^2 + 112x
\]

---

### 4. Find and simplify \( \frac{h(x)}{g(x)} \)

We are given:
- \( g(x) = x + 7 \)
- \( h(x) = 3x^2 + 23x + 14 \)

To find \( \frac{h(x)}{g(x)} \), we need to factor \( h(x) \) and check if any common factors can be canceled.

First, factor \( h(x) = 3x^2 + 23x + 14 \):

Find two numbers that multiply to \( 3 \times 14 = 42 \) and add to 23. These numbers are 21 and 2.

So, we can factor \( h(x) \) as follows:

\[
h(x) = 3x^2 + 21x + 2x + 14
\]
\[
h(x) = 3x(x + 7) + 2(x + 7)
\]
\[
h(x) = (3x + 2)(x + 7)
\]

Now, simplify \( \frac{h(x)}{g(x)} \):

\[
\frac{h(x)}{g(x)} = \frac{(3x + 2)(x + 7)}{x + 7}
\]

Cancel the common factor \( (x + 7) \):

\[
\frac{h(x)}{g(x)} = 3x + 2
\]

So, the simplified expression is:

\[
\frac{h(x)}{g(x)} = 3x + 2
\]

---

### 5. Domain restriction for \( \frac{g(x)}{f(x)} \)

We are given:
- \( g(x) = x + 7 \)
- \( f(x) = 8x \)

To find the domain restriction for \( \frac{g(x)}{f(x)} \), remember that division by zero is undefined. Set \( f(x) = 0 \) and solve for \( x \):

\[
8x = 0
\]
\[
x = 0
\]

So, the domain restriction is \( x \neq 0 \).

---

### Summary:

1. \( g(x) + h(x) = 3x^2 + 24x + 21 \)
2. \( h(x) - g(x) = 3x^2 + 22x + 7 \)
3. \( f(x) \times h(x) = 24x^3 + 184x^2 + 112x \)
4. \( \frac{h(x)}{g(x)} = 3x + 2 \)
5. Domain restriction for \( \frac{g(x)}{f(x)} \): \( x \neq 0 \)

Would you like further explanation on any of the steps?

### Related Questions:
1. How would the results change if \( f(x) = 8x^2 \)?
2. What is the derivative of \( g(x) + h(x) \)?
3. How would you factor \( 3x^2 + 24x + 21 \)?
4. What is the range of \( \frac{h(x)}{g(x)} \)?
5. How does the product of two functions affect their domains?

### Tip:
When factoring quadratic expressions, always check if there's a common factor before applying other factoring techniques.

Given the functions: f(x) = 8x, g(x) = x + 7, h(x) = 3x^2 + 23x + 14. Determine each of the following: 1) g(x) + h(x), 2) h(x) - g(x), 3) f(x) × h(x), 4) h(x) / g(x), and 5) Domain restriction for g(x) / f(x).

Solve polynomial problems involving the functions f(x) = 8x, g(x) = x + 7, and h(x) = 3x^2 + 23x + 14. This tutorial covers polynomial addition, subtraction, multiplication, factoring, and finding domain restrictions for rational expressions. Suitable for Grades 9-10 students.

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