Question: Joe is building a post for his mailbox. To find the correct dimensions, he needs to expand this expression: $$(x - 3)(x - 9)(x - 4)$$. Select the equivalent expression written in the format $ax^3 + bx^2 + cx + d$.
a.) $x^3 - 16x^2 + 75x - 108$
b.) $x^3 + 10x^2 - 3x - 108$
c.) $x^3 + 16x^2 - 75x + 108$
d.) $x^3 - 10x^2 - 3x + 108$
1. **State the problem:** We need to expand the product $$(x - 3)(x - 9)(x - 4)$$ and write it in the form $ax^3 + bx^2 + cx + d$.
2. **Use the distributive property (FOIL for binomials) step-by-step:**
First, multiply the first two binomials:
$$
(x - 3)(x - 9) = x^2 - 9x - 3x + 27 = x^2 - 12x + 27
$$
3. **Now multiply the result by the third binomial $(x - 4)$:**
$$
(x^2 - 12x + 27)(x - 4)
$$
Apply distributive property:
$$
= x^2(x - 4) - 12x(x - 4) + 27(x - 4)
$$
4. **Multiply each term:**
$$
= x^3 - 4x^2 - 12x^2 + 48x + 27x - 108
$$
5. **Combine like terms:**
$$
x^3 - (4x^2 + 12x^2) + (48x + 27x) - 108 = x^3 - 16x^2 + 75x - 108
$$
6. **Final answer:** The expanded form is
$$
\boxed{x^3 - 16x^2 + 75x - 108}
$$
This matches option a.)