Given that f, of, x, equals, x, squared, plus, 6, x, minus, 16f(x)=x 
2
 +6x−16 and g, of, x, equals, x, plus, 8g(x)=x+8, find f, of, x, dot, g, of, xf(x)⋅g(x) and express the result as a polynomial in simplest form.

To find \( f(x) \cdot g(x) \), we need to multiply the two given functions \( f(x) \) and \( g(x) \).

### Given:
- \( f(x) = x^2 + 6x - 16 \)
- \( g(x) = x + 8 \)

Now, calculate \( f(x) \cdot g(x) \):

\[
f(x) \cdot g(x) = (x^2 + 6x - 16)(x + 8)
\]

### Step-by-step multiplication:
1. Distribute each term in \( f(x) \) by each term in \( g(x) \).

\[
(x^2 + 6x - 16)(x + 8) = x^2(x + 8) + 6x(x + 8) - 16(x + 8)
\]

2. Multiply out each part:
\[
x^2(x + 8) = x^3 + 8x^2
\]
\[
6x(x + 8) = 6x^2 + 48x
\]
\[
-16(x + 8) = -16x - 128
\]

3. Combine all the terms:
\[
f(x) \cdot g(x) = x^3 + 8x^2 + 6x^2 + 48x - 16x - 128
\]

4. Combine like terms:
\[
f(x) \cdot g(x) = x^3 + (8x^2 + 6x^2) + (48x - 16x) - 128
\]
\[
f(x) \cdot g(x) = x^3 + 14x^2 + 32x - 128
\]

### Final answer:
\[
f(x) \cdot g(x) = x^3 + 14x^2 + 32x - 128
\]

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the degree of the polynomial \( f(x) \cdot g(x) \)?
2. How would you factor the polynomial \( f(x) \cdot g(x) \)?
3. Can you find \( f(2) \cdot g(2) \) directly from the given functions?
4. How would the product change if \( g(x) = x - 8 \) instead?
5. What is the remainder when \( f(x) \cdot g(x) \) is divided by \( x - 2 \)?

**Tip:** When multiplying polynomials, keep track of the degrees of each term to ensure that you combine like terms correctly.

Given that f(x) = x^2 + 6x - 16 and g(x) = x + 8, find f(x)⋅g(x) and express the result as a polynomial in simplest form.

Learn how to multiply the polynomials f(x) = x^2 + 6x - 16 and g(x) = x + 8. This problem covers key algebraic concepts like polynomial multiplication with a detailed step-by-step solution. Suitable for students in Grades 9-11.

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