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The profit of a retail store is the amount of money that remains after all expenses have been subtracted from the total revenue. In our exercise, the store manager has created a formula to determine the monthly profit, given a specific amount spent on advertising. The formula is: \[ P = x^3 - 20x^2 + 100x \] Here, \(P\) represents the profit in thousands of dollars, and \(x\) is the advertising expense in tens of thousands of dollars. By understanding this relationship, the manager can make informed decisions about how much to spend on advertising to achieve desired profit levels.

Advertising expense is the amount of money spent on promoting a store's products or services. This can include costs for online ads, print ads, billboards, and more. In our problem, the advertising expense is represented by the variable \(x\). \ Spending on advertising can help increase a store’s revenue by attracting more customers and increasing sales. However, spending too much might not always lead to higher profits if the costs outweigh the benefits.
This is why it is crucial to find the optimal advertising expense that maximizes profit. The optimal value of \(x\) ensures that every dollar spent contributes effectively to the store's profitability.

Profit maximization refers to finding the level of output or input that leads to the highest possible profit. In the context of our exercise, it involves determining the best amount of advertising expense (\(x\)) to maximize monthly profit (\(P\)).
To find the optimal value, you may need to solve mathematical equations that represent the relationships between different variables. Here, the profit formula is a cubic equation, \[ x^3 - 20x^2 + 100x - 147 = 0 \] We solve this to find the value of \(x\) that yields a profit (\(P\)) of \$147,000. This process involves substituting \(P = 147\) into the equation and solving for \(x\).

Solving cubic equations like ours isn't always straightforward. Analytical solutions can be complex, so we often turn to numerical methods to find an approximate solution. Numerical methods include techniques like Newton's method, bisection method, and using graphing calculators or software.
These methods involve iteratively guessing and checking values of \(x\) until you find one that makes the equation true. For example, in our exercise, we use a numerical method to find the root of the equation \[ x^3 - 20x^2 + 100x - 147 = 0 \] This root, which is approximately \(x = 3\), helps us determine the advertising expense needed to achieve a \$147,000 profit.\ To verify, we substitute \(x = 3\) back into the original profit equation and check if the result matches the expected profit.