Problem 15 - Algebra and Trigonometry with Analytic Geometry

Question

Find the zeros of \(f\). $$ f(x)=x^{3}\left(4 e^{4 x}\right)+3 x^{2} e^{4 x} $$

Step-by-Step Solution

Answer

The zeros of the function are \(x = 0\) and \(x = -\frac{3}{4}\).

Step 1: Factor Out Common Terms

Start by examining the function \[ f(x) = x^{3} (4 e^{4x}) + 3x^{2} e^{4x} \]Both terms contain \(x^2 e^{4x}\) as a common factor. Factor it out:\[f(x) = x^2 e^{4x} (4x + 3)\]

Step 2: Set Factorized Function to Zero

To find the zeros, set the factorized function equal to zero:\[x^2 e^{4x} (4x + 3) = 0\]This equals zero if any factor equals zero.

Step 3: Solve Each Factor Separately

For the expression to be zero, we consider the zeroes of each factor separately. The factors are:1. \(x^2 = 0\) which yields \(x = 0\).2. \(e^{4x} = 0\), which can never be zero since the exponential function doesn't cross the x-axis.3. \(4x + 3 = 0\), solve for \(x\):\[4x + 3 = 0 \x = -\frac{3}{4}\]Thus, the solutions to \(f(x) = 0\) are \(x = 0\) and \(x = -\frac{3}{4}\).

Key Concepts

Polynomial equations

Polynomials are mathematical expressions that involve sums of powers in one or more variables multiplied by coefficients. Understanding polynomial equations is crucial for solving various mathematical problems. In polynomial equations, each term is a product of a constant and a power of the variable. The powers, or exponents, are whole numbers, giving polynomial functions their distinct form.

When dealing with polynomial equations, the degree of the polynomial tells you the highest power of the variable. For example, in our function, the term \(x^3\) indicates a third-degree polynomial, which suggests it could have up to three real roots. However, not all terms may contribute to unique zeros. It's essential to effectively manage the polynomials by factoring, **identifying common terms**, and simplifying expressions to find these zeros.

• Identify the degree of the polynomial: Highest power in the function.
• Understand that each power could contribute a zero.
• Factor the polynomial when possible to simplify solving.

By examining the nature of polynomial equations, we make solving them more digestible, ensuring we find meaningful solutions.

Exponential functions

Exponential functions are distinct from polynomial functions as they involve variables in the exponent, like the term \(e^{4x}\) in our exercise. These functions have a standard form of \(a^x\), where \(a\) is a constant and \(x\) is the exponent. In our case, \(e\) is the base, known as Euler's number (approximately 2.718). Exponential functions exhibit constant growth rates, meaning they increase rapidly as the value of \(x\) becomes large.

Characteristics of exponential functions include:

• They never equal zero.
• They have a horizontal asymptote at \(y = 0\).
• They grow quickly and have significant applications in real-life scenarios, such as compound interest in finance or population growth models.

Understanding the behavior of exponential functions is crucial when solving equations involving them, as it shapes our approach to setting them equal to zero and identifying solutions in combined forms with other functions, like polynomials in our example.

Factoring expressions

Factoring is an invaluable technique in algebra that involves breaking down complex expressions into simpler components termed factors. This method helps in solving equations and simplifies expressions, making them easier to handle.

The concept of factoring revolves around identifying common elements or patterns within an equation to rewrite it in a more manageable form. In our function \(f(x) = x^3 \cdot 4e^{4x} + 3x^2 \cdot e^{4x}\), by recognizing \(x^2 \cdot e^{4x}\) as a common factor, we could reduce the expression to \(x^2 \cdot e^{4x} \cdot (4x + 3)\). This simplification reveals the zeros via easier equations to solve.

• Identify common factors to factor out.
• Re-express the equation using these common elements.
• Simplify further where possible to facilitate solving the equation.

By using factoring effectively, we can uncover the solutions to the polynomial when simpler methods of direct solving aren't straightforward or visible.