How do you find the product of the functions f(x) = 27x^5 + 33x^4 + 21x^3 and g(x) = 3x^2?

To find the product of the two functions, f(x) and g(x), we begin with their definitions:

• f(x) = 27x^5 + 33x^4 + 21x^3
• g(x) = 3x^2

The product of the functions, denoted as (f * g)(x), can be computed by multiplying each term in f(x) by the entire function g(x).

Let’s perform the multiplication step-by-step:

1. Multiply the leading term of f(x):

27x^5 * 3x^2 = 81x^{7}

2. Multiply the second term of f(x):

33x^4 * 3x^2 = 99x^{6}

3. Multiply the third term of f(x):

21x^3 * 3x^2 = 63x^{5}

Now that we have the products of each term, we can combine them:

(f * g)(x) = 81x^{7} + 99x^{6} + 63x^{5}

Therefore, the product of the functions is:

(f * g)(x) = 81x^{7} + 99x^{6} + 63x^{5}

And that’s how you compute the product of f(x) and g(x)!