Given that f, of, x, equals, x, squared, minus, 17, x, plus, 72f(x)=x 
2
 −17x+72 and g, of, x, equals, x, minus, 8g(x)=x−8, find left parenthesis, f, dot, g, right parenthesis, left parenthesis, x, right parenthesis(f⋅g)(x) and express the result as a polynomial in simplest form.

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

Given:
\[
f(x) = x^2 - 17x + 72
\]
\[
g(x) = x - 8
\]

The product \( (f \cdot g)(x) \) is defined as:
\[
(f \cdot g)(x) = f(x) \cdot g(x)
\]

This means we multiply the two functions:
\[
(f \cdot g)(x) = (x^2 - 17x + 72)(x - 8)
\]

Now, apply the distributive property (also known as FOIL for binomials):

\[
(x^2 - 17x + 72)(x - 8) = x^2(x - 8) - 17x(x - 8) + 72(x - 8)
\]

Distribute each term:

\[
= x^3 - 8x^2 - 17x^2 + 136x + 72x - 576
\]

Now, combine like terms:

\[
= x^3 - 25x^2 + 208x - 576
\]

Thus, the simplified polynomial is:

\[
(f \cdot g)(x) = x^3 - 25x^2 + 208x - 576
\]

Let me know if you need more details or have any questions!

### Related Questions:
1. What is the degree of the polynomial \( f(x) \cdot g(x) \)?
2. How do you find the roots of the polynomial \( f(x) \cdot g(x) \)?
3. What happens if you substitute \( x = 0 \) in \( (f \cdot g)(x) \)?
4. How does the polynomial change if \( g(x) = x - 7 \) instead of \( x - 8 \)?
5. Can you express \( f(x) \) and \( g(x) \) as factors of \( f \cdot g(x) \)?

### Tip:
Always combine like terms carefully when multiplying polynomials to avoid mistakes in simplification.

Given that f(x) = x^2 - 17x + 72 and g(x) = x - 8, find (f ⋅ g)(x) and express the result as a polynomial in simplest form.

Learn how to multiply the polynomials f(x) = x^2 - 17x + 72 and g(x) = x - 8. This step-by-step solution simplifies the polynomial expression using the distributive property. Perfect for students in Grades 9-11.

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