https://file.aiquickdraw.com/mathbot/file/1725427801959q9635u9d.jpg

https://file.aiquickdraw.com/mathbot/file/1725427848590a6k36pq2.jpg

The problem you provided asks to solve the following, given that \( F(x) = x^2 + 5x + 6 \) and \( H(x) = x + 2 \):

1. Find \( (F \cdot H)(x) \)
2. Find \( (F \div H)(x) \)
3. Find \( (F + H)(x) \)

Let's solve each part step by step.

### 1. \( (F \cdot H)(x) \)
To find \( (F \cdot H)(x) \), multiply the functions \( F(x) \) and \( H(x) \):

\[
(F \cdot H)(x) = F(x) \cdot H(x) = (x^2 + 5x + 6) \cdot (x + 2)
\]

Expand the expression:

\[
(F \cdot H)(x) = x^2 \cdot x + x^2 \cdot 2 + 5x \cdot x + 5x \cdot 2 + 6 \cdot x + 6 \cdot 2
\]

Simplifying:

\[
(F \cdot H)(x) = x^3 + 2x^2 + 5x^2 + 10x + 6x + 12 = x^3 + 7x^2 + 16x + 12
\]

So:

\[
(F \cdot H)(x) = x^3 + 7x^2 + 16x + 12
\]

### 2. \( (F \div H)(x) \)
To find \( (F \div H)(x) \), divide \( F(x) \) by \( H(x) \):

\[
(F \div H)(x) = \frac{F(x)}{H(x)} = \frac{x^2 + 5x + 6}{x + 2}
\]

Factor the numerator:

\[
F(x) = x^2 + 5x + 6 = (x + 2)(x + 3)
\]

Now, divide:

\[
(F \div H)(x) = \frac{(x + 2)(x + 3)}{x + 2}
\]

Since \( x + 2 \) cancels out, we have:

\[
(F \div H)(x) = x + 3
\]

### 3. \( (F + H)(x) \)
To find \( (F + H)(x) \), add the functions \( F(x) \) and \( H(x) \):

\[
(F + H)(x) = F(x) + H(x) = (x^2 + 5x + 6) + (x + 2)
\]

Combine like terms:

\[
(F + H)(x) = x^2 + 6x + 8
\]

### Summary of Results
1. \( (F \cdot H)(x) = x^3 + 7x^2 + 16x + 12 \)
2. \( (F \div H)(x) = x + 3 \)
3. \( (F + H)(x) = x^2 + 6x + 8 \)

Would you like any further details or have any questions? Here are five related questions to expand your understanding:

1. What would be \( (F - H)(x) \) given the same functions?
2. How can the division \( (F \div H)(x) \) change if \( H(x) \) were different?
3. What is the derivative of \( (F \cdot H)(x) \)?
4. What would be the roots of \( (F + H)(x) \)?
5. How does the degree of the polynomial change in \( (F \cdot H)(x) \) compared to \( F(x) \) and \( H(x) \) individually?

**Tip:** When working with polynomial expressions, always consider factoring and simplifying to make your calculations easier.

Given F(x) = x² + 5x + 6, and H(x) = x + 2. Find (F · H)(x), (F ÷ H)(x), and (F + H)(x).

Learn how to solve problems involving operations on polynomial functions, including multiplication, division, and addition. This problem explores the functions F(x) = x² + 5x + 6 and H(x) = x + 2. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation