Problem 29 - Algebra and Trigonometry

Question

Solve the equation. $$ 4 x^{3} e^{-3 x}-3 x^{4} e^{-3 x}=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = \frac{4}{3}\).

Step 1: Factor the Equation

The given equation is \(4x^3 e^{-3x} - 3x^4 e^{-3x} = 0\). We can factor out the common term \(x^3 e^{-3x}\), which simplifies the equation to \(x^3 e^{-3x} (4 - 3x) = 0\).

Step 2: Apply the Zero Product Property

According to the Zero Product Property, if a product of factors is zero, at least one of the factors must be zero. Set each factor equal to zero: \(x^3 e^{-3x} = 0\) and \(4 - 3x = 0\).

Step 3: Solve for \(x\) from the First Factor

The factor \(x^3 e^{-3x} = 0\) implies that either \(x^3 = 0\) or \(e^{-3x} = 0\). Since the exponential function \(e^{-3x}\) is never zero, we only consider \(x^3 = 0\), which gives the solution \(x = 0\).

Step 4: Solve for \(x\) from the Second Factor

The equation \(4 - 3x = 0\) can be simplified by solving for \(x\). Subtract 4 from both sides to get \(-3x = -4\), then divide both sides by \(-3\) to find \(x = \frac{4}{3}\).

Key Concepts

Factorization

Understanding factorization in algebra is like finding the building blocks of a complex expression. It involves breaking down an expression into simpler elements called factors, which when multiplied together give back the original expression.

In our problem, the expression given is:

• \(4x^3 e^{-3x} - 3x^4 e^{-3x} = 0\)

This looks a bit intimidating at first glance, but fear not! Just like how cookies crumble, you can break this down. Notice that both terms on the left side of the equation share common parts: \(x^3 e^{-3x}\). By factoring these out, you simplify the equation to:

• \(x^3 e^{-3x} (4 - 3x) = 0\)

Factorizing makes solving the equation easier and reveals more about the structure of the expression by unearthing common parts. This process sets us up perfectly for using the Zero Product Property next.

Zero Product Property

The Zero Product Property is a fantastic logical rule that helps you solve equations like a pro!

When you have several factors multiplied together equating to zero, one (or more) of those factors must also be zero.Our factorized equation from before is:

• \(x^3 e^{-3x} (4 - 3x) = 0\)

According to the Zero Product Property, you can split this into:

• \(x^3 e^{-3x} = 0\)
• \(4 - 3x = 0\)

This helps you to tackle sometimes complicated equations and break them down into simpler, more manageable bites. It's like having multiple roads that can lead to the correct solution, as each part of the equation has its own method for finding the answer, making your problem-solving journey a lot more straightforward.

Exponential function

Exponential functions can seem a bit mysterious, but they're powerful and incredibly useful in mathematics. An exponential function is expressed as \(e^x\) and represents constant growth or decay.

In our equation, it is part of the term \(e^{-3x}\).A key property to note is that the exponential function \(e^x\) never equals zero for any real number \(x\). This fact simplifies solving because if you try to solve:

• \(e^{-3x} = 0\)

You learn quickly it's not possible. The revelation here is simple but crucial in focusing our efforts on other parts of the equation. This information redirects your attention to solving easier parts, like \(x^3 = 0\), sparing you from fruitless calculations.

Polynomial roots

Solving for polynomial roots involves finding the values of \(x\) that make the polynomial zero. The roots can tell us a lot about the polynomial function, like where it intersects with the x-axis.

From our original factorized equation, we have:

• \(x^3 = 0\): The only root here is \(x = 0\).
• \(4 - 3x = 0\): Simplifies to \(x = \frac{4}{3}\).

These are the roots of our polynomial, which we found by using factorization and the Zero Product Property.Polynomial roots are crucial as they provide solutions and insights into the behavior of polynomial equations. Detecting these values is akin to identifying the key points where the polynomial graph crosses or touches the x-axis. This can be incredibly valuable in multiple areas of science and engineering.