Calculate the Product Representation of f(x) = x² - 2x - 3

Find the representation of the product of the following function

$f(x)=x^2-2x-3$

The problem requires finding the product representation of the quadratic function $f(x) = x^2 - 2x - 3$.

Let's execute the factorization of the quadratic equation:

• The standard form for the function is $f(x) = ax^2 + bx + c$. Here, $a = 1$, $b = -2$, $c = -3$.
• We seek two numbers that multiply to $c = -3$ and sum to $b = -2$.
• Checking possible integer pairs: $(-3, 1)$ can accomplish this, since $-3 \times 1 = -3$ and $-3 + 1 = -2$.
• The factorization becomes $f(x) = (x - 3)(x + 1)$.

To verify, we can expand the binomials:

$(x - 3)(x + 1) = x^2 + x - 3x - 3 = x^2 - 2x - 3$.

This matches the original polynomial, confirming the product representation is correct.

In conclusion, the factorization or product representation of the given quadratic function is $\mathbf{(x-3)(x+1)}$.