A. Find (fg)(x) for each pair of functions. (12 points)

1. f(x) = 2x + 6 and g(x) = 3x - 3

Solution and answer:

2. f(x) = 3x and g(x) = x ^ 2 - 2x + 1

Solution and answer:

3. f(x) = 3x + 6 and g(x) = 2x ^ 2 + x + 4

Solution and answer:

4. f(x) = x ^ 2 - 3x + 2 and g(x) = x ^ 2 - 2x + 1

Solution and answer:

To find \((fg)(x)\) for each pair of functions, you multiply the two functions together, which means \((fg)(x) = f(x) \cdot g(x)\). Below are the solutions and answers for each pair of functions:

### 1. \(f(x) = 2x + 6\) and \(g(x) = 3x - 3\)

**Solution:**

\[
(fg)(x) = f(x) \cdot g(x) = (2x + 6) \cdot (3x - 3)
\]

Expand the product using the distributive property:

\[
(fg)(x) = 2x \cdot 3x + 2x \cdot (-3) + 6 \cdot 3x + 6 \cdot (-3)
\]

\[
(fg)(x) = 6x^2 - 6x + 18x - 18
\]

Combine like terms:

\[
(fg)(x) = 6x^2 + 12x - 18
\]

**Answer:** \((fg)(x) = 6x^2 + 12x - 18\)

---

### 2. \(f(x) = 3x\) and \(g(x) = x^2 - 2x + 1\)

**Solution:**

\[
(fg)(x) = f(x) \cdot g(x) = 3x \cdot (x^2 - 2x + 1)
\]

Distribute \(3x\) across each term in \(g(x)\):

\[
(fg)(x) = 3x \cdot x^2 + 3x \cdot (-2x) + 3x \cdot 1
\]

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

**Answer:** \((fg)(x) = 3x^3 - 6x^2 + 3x\)

---

### 3. \(f(x) = 3x + 6\) and \(g(x) = 2x^2 + x + 4\)

**Solution:**

\[
(fg)(x) = f(x) \cdot g(x) = (3x + 6) \cdot (2x^2 + x + 4)
\]

Expand the product:

\[
(fg)(x) = 3x \cdot 2x^2 + 3x \cdot x + 3x \cdot 4 + 6 \cdot 2x^2 + 6 \cdot x + 6 \cdot 4
\]

\[
(fg)(x) = 6x^3 + 3x^2 + 12x + 12x^2 + 6x + 24
\]

Combine like terms:

\[
(fg)(x) = 6x^3 + 15x^2 + 18x + 24
\]

**Answer:** \((fg)(x) = 6x^3 + 15x^2 + 18x + 24\)

---

### 4. \(f(x) = x^2 - 3x + 2\) and \(g(x) = x^2 - 2x + 1\)

**Solution:**

\[
(fg)(x) = f(x) \cdot g(x) = (x^2 - 3x + 2) \cdot (x^2 - 2x + 1)
\]

Expand the product:

\[
(fg)(x) = x^2 \cdot (x^2 - 2x + 1) - 3x \cdot (x^2 - 2x + 1) + 2 \cdot (x^2 - 2x + 1)
\]

Now expand each term:

\[
(fg)(x) = (x^4 - 2x^3 + x^2) - (3x^3 - 6x^2 + 3x) + (2x^2 - 4x + 2)
\]

Combine like terms:

\[
(fg)(x) = x^4 - 5x^3 + 9x^2 - 7x + 2
\]

**Answer:** \((fg)(x) = x^4 - 5x^3 + 9x^2 - 7x + 2\)

---

### Questions to Expand Understanding:

1. How would the results change if any of the functions were negative?
2. What happens when one of the functions is a constant? How does it affect the product?
3. What does the degree of the resulting polynomial tell you about the functions involved?
4. Is it possible for the product of two non-zero functions to be zero at certain points? Explain with an example.
5. How would you approach the problem if one of the functions were a rational function instead of a polynomial?

### Tip:

When multiplying polynomials, focus on systematically expanding and combining like terms to avoid mistakes.

Explore comprehensive solutions to multiply pairs of functions f(x) and g(x). This mathematical problem involves multiplying various pairs of functions, including linear and quadratic functions, providing step-by-step solutions and answers. Suitable for high school students.

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