Discovering the Product Representation of f(x) = x^2 - 7x + 12

To solve the problem of finding the product (factored) representation of the quadratic function $f(x) = x^2 - 7x + 12$, we proceed as follows:

• Step 1: Identify the function, which is $f(x) = x^2 - 7x + 12$.
• Step 2: We need to factor this quadratic expression. We're looking for two numbers whose product is 12 and whose sum is -7.
• Step 3: The factor pairs of 12 are $(1, 12)$, $(2, 6)$, $(3, 4)$, including negative pairs because the sum must be negative.
• Step 4: Consider the pair $(-3, -4)$. The product $(-3) \times (-4)$ equals 12, and the sum $(-3) + (-4)$ equals -7.

Therefore, the factors of the quadratic expression are $x - 3$ and $x - 4$. This implies that the function $f(x)$ can be expressed in product form as:

$f(x) = (x - 3)(x - 4)$

This means the correct factorization is $(x - 3)(x - 4)$, which corresponds to choice 3 from the given options.

Thus, the representation of the product of the function is $(x - 3)(x - 4)$.