- Start by writing the product as \((f \cdot g)(x) = (5x + 3)(x^2 - 4x)\).
- Expand the expression using the distributive property: \(5x \cdot x^2 + 5x \cdot (-4x) + 3 \cdot x^2 + 3 \cdot (-4x) = 5x^3 - 20x^2 + 3x^2 - 12x\).
- Simplify the expression by combining the like terms \(-20x^2\) and \(3x^2\).
- State the final polynomial form: \((f \cdot g)(x) = \boxed{5x^3 - 17x^2 - 12x}\).
Explanation
- Analysis of the Problem
We are given two functions \(f(x) = 5x + 3\) and \(g(x) = x^2 - 4x\). The goal is to find their product, denoted by \((f \cdot g)(x)\), and express it as a simplified polynomial. By definition, the product function \((f \cdot g)(x)\) is the point-wise product of the two functions, which means \((f \cdot g)(x) = f(x) \cdot g(x)\). Substituting the given expressions, we have:
\[(f \cdot g)(x) = (5x + 3)(x^2 - 4x)\]
- Expansion of the Product
To find the product, we use the distributive property to multiply every term in the first binomial by every term in the second. We expand the expression \((5x + 3)(x^2 - 4x)\) as follows:
\[(5x + 3)(x^2 - 4x) = (5x)(x^2) + (5x)(-4x) + (3)(x^2) + (3)(-4x)\]
Calculating each product individually:
- \((5x)(x^2) = 5x^3\)
- \((5x)(-4x) = -20x^2\)
- \((3)(x^2) = 3x^2\)
- \((3)(-4x) = -12x\)
Thus, the expression becomes:
\[5x^3 - 20x^2 + 3x^2 - 12x\]
- Simplification and Final Result
Now we combine the like terms, which are \(-20x^2\) and \(3x^2\):
\[-20x^2 + 3x^2 = -17x^2\]
Combining all the terms, we get the final polynomial:
\[(f \cdot g)(x) = 5x^3 - 17x^2 - 12x\]
The simplified form of the product function is \(\boxed{5x^3 - 17x^2 - 12x}\).
Examples
In economics and business, if \(f(x)\) represents the number of units sold of a product at price \(x\), and \(g(x)\) represents the profit margin per unit at price \(x\), the total profit function is often the product of these two functions. By finding the product function \((f \cdot g)(x)\), business analysts can determine how the total profit behaves as a polynomial function of the price \(x\), allowing them to calculate the price that maximizes profit through calculus techniques.