Question

If m(x) = x2 + 3 and n(x) = 5x + 9, which expression is equivalent to (mn)(x)?
5x3 + 9x2 + 15x + 27
25x2 + 90x + 84
x2 + 5x + 12
5x2 + 24

Answer 1

To find \((mn)(x)\), we need to first compute the product of the functions \(m(x)\) and \(n(x)\):

1. Identify the functions:
\[
m(x) = x^2 + 3
\]
\[
n(x) = 5x + 9
\]

2. Compute \((mn)(x) = m(x) \cdot n(x)\):
\[
(mn)(x) = (x^2 + 3)(5x + 9)
\]

3. Use the distributive property (also known as the FOIL method for binomials) to expand the product:
\[
= x^2 \cdot 5x + x^2 \cdot 9 + 3 \cdot 5x + 3 \cdot 9
\]
\[
= 5x^3 + 9x^2 + 15x + 27
\]

So, we find that \((mn)(x) = 5x^3 + 9x^2 + 15x + 27\).

Now let's compare it with the given expressions:

1. **5x^3 + 9x^2 + 15x + 27** (This is our result)
2. 25x^2 + 90x + 84
3. x^2 + 5x + 12
4. 5x^2 + 24

The expression equivalent to \((mn)(x)\) is:
\[
\boxed{5x^3 + 9x^2 + 15x + 27}
\]