Determining if $x - 4$ is a Factor of $f(x) = x^3 - 64$ Using the Factor Theorem

Determining if $x - 5$ is a Factor of $f(x) = x^3 - 125$ Using the Factor Theorem

Determining if $x - 6$ is a Factor of $f(x) = x^3 - 216$ Using the Factor Theorem

A chemical engineer uses a polynomial function $C(x)$ to model the concentration of a substance in a reaction at time $x$. If the engineer observes that the concentration is zero at $x = 12$ seconds (meaning $C(12) = 0$), which expression must be a factor of the polynomial $C(x)$ according to the Factor Theorem?

An analyst uses a polynomial function $R(x)$ to model the revenue of a product based on its price $x$. If the analyst finds that the revenue is zero when the price is 15 dollars (meaning $R(15) = 0$), the Factor Theorem states that $(x + 15)$ is a factor of the revenue function.

Factor Theorem in Structural Modeling

In your role as a business analyst, you are using polynomial functions to model sales data and identify break-even points where profit is zero. Match each observed data point or algebraic property of the profit model $f(x)$ with its equivalent implication according to the Factor Theorem.

You are an inventory analyst using a polynomial model, $I(t)$, to forecast warehouse stock levels over time $t$. If you determine that the inventory hits exactly zero at a specific time $c$ (meaning $I(c) = 0$), the Factor Theorem guarantees that the binomial expression $(t - c)$ must be a ____ of the polynomial function $I(t)$.

Communicating the Factor Theorem in Business Modeling

As an assistant analyst at a shipping firm, you are helping verify a cost-optimization model represented by a polynomial function $C(x)$. Your supervisor wants you to use the Factor Theorem to determine whether a potential cost-reduction factor $x - 5$ is an exact factor of $C(x)$ (which would mean a 5-unit reduction results in a net cost change of zero). Arrange the steps you must take to verify this condition in the correct chronological order, from first to last.