Zeros Of H(x) = 3x^3 + 9x^2 + 12x: A Step-by-Step Guide

In mathematics, finding the zeros of a function is a fundamental concept with wide-ranging applications. The zeros, also known as roots or x-intercepts, are the values of x for which the function h(x) equals zero. In this detailed guide, we will walk through the process of finding the zeros of the function h(x) = 3x³ + 9x² + 12x. This is a cubic polynomial, and understanding how to solve for its zeros is crucial for various mathematical and real-world problems. This comprehensive guide aims to provide you with a clear, step-by-step approach to solving such problems.

Understanding Zeros of a Function

Before diving into the specifics of our function, let's clarify what we mean by the zeros of a function. In simple terms, the zeros are the x-values that make the function equal to zero. Graphically, these are the points where the function's graph intersects the x-axis. Finding these zeros is essential because they often represent critical points in a system, such as equilibrium points in physics or break-even points in economics. Understanding how to find zeros is a cornerstone of algebra and calculus, providing a foundation for more advanced mathematical concepts.

For a polynomial function like h(x) = 3x³ + 9x² + 12x, the zeros tell us where the graph of the polynomial crosses the x-axis. These points are crucial for understanding the behavior of the function, including its increasing and decreasing intervals, local maxima and minima, and overall shape. Moreover, finding the zeros is often the first step in solving more complex problems, such as optimization problems or curve sketching. So, let's get started on how to find these crucial values for our function.

Step 1: Factoring out the Common Factor

Our function is h(x) = 3x³ + 9x² + 12x. The first step in finding the zeros is to simplify the function by factoring out the greatest common factor (GCF). Looking at the terms, we can see that each term has a common factor of 3x. Factoring this out simplifies the equation and makes it easier to work with. Factoring is a crucial technique in algebra, allowing us to break down complex expressions into simpler components. This step not only makes the equation more manageable but also reveals key information about the function's behavior.

So, let's factor out 3x from the function:

h(x) = 3x(x² + 3x + 4)

Now, we have expressed the function as a product of 3x and a quadratic expression (x² + 3x + 4). This is a significant step because it allows us to use the zero-product property, which states that if the product of factors is zero, then at least one of the factors must be zero. Thus, we can set each factor equal to zero and solve for x. This simplifies our task into solving a simpler linear equation and a quadratic equation.

Step 2: Setting the Factors to Zero

Now that we have h(x) = 3x(x² + 3x + 4), we can apply the zero-product property. This property is a cornerstone of solving polynomial equations and is incredibly useful when an equation is factored. According to the zero-product property, if a * b = 0*, then either a = 0 or b = 0 (or both). This allows us to break down a complex equation into simpler equations that are easier to solve.

So, we set each factor equal to zero:

1. 3x = 0
2. x² + 3x + 4 = 0

The first equation, 3x = 0, is a simple linear equation that can be easily solved by dividing both sides by 3. The second equation, x² + 3x + 4 = 0, is a quadratic equation that requires a different approach. We will address each of these equations separately in the following steps, starting with the simpler linear equation.

Step 3: Solving the Linear Equation

We have the linear equation 3x = 0. To solve for x, we simply divide both sides of the equation by 3: This is a basic algebraic manipulation, but it's crucial for isolating the variable and finding its value. Solving linear equations is a fundamental skill in algebra, and it often serves as a building block for solving more complex equations.

(3x) / 3 = 0 / 3

This simplifies to:

x = 0

So, one of the zeros of the function h(x) is x = 0. This means that the graph of the function h(x) intersects the x-axis at the point (0, 0). This is a significant piece of information, but we still need to find the zeros from the quadratic factor. Let's move on to solving the quadratic equation.

Step 4: Solving the Quadratic Equation

Now we need to solve the quadratic equation x² + 3x + 4 = 0. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, the quadratic expression does not factor easily, so we will use the quadratic formula. The quadratic formula is a powerful tool that can solve any quadratic equation, regardless of whether it can be factored.

The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / (2a)

For our equation, x² + 3x + 4 = 0, we have a = 1, b = 3, and c = 4. Plugging these values into the quadratic formula, we get:

x = (-3 ± √(3² - 4(1)(4))) / (2(1))

Let's simplify this expression step by step.

Step 5: Simplifying the Quadratic Formula

We have the expression:

x = (-3 ± √(3² - 4(1)(4))) / (2(1))

First, let's simplify the expression inside the square root:

3² - 4(1)(4) = 9 - 16 = -7

So, the expression becomes:

x = (-3 ± √(-7)) / 2

Since we have a negative number inside the square root, the solutions will be complex numbers. The square root of -7 can be written as √(-1) * √7 = i√7, where i is the imaginary unit (i² = -1). This is a key step in dealing with quadratic equations that have no real roots. Complex numbers extend the real number system and are crucial in many areas of mathematics and physics.

Now, we can rewrite the expression as:

x = (-3 ± i√7) / 2

This gives us two complex solutions.

Step 6: Identifying the Zeros

From the quadratic formula, we have two complex solutions:

x = (-3 + i√7) / 2 x = (-3 - i√7) / 2

These are the two complex zeros of the function h(x). Combining these with the real zero we found earlier, x = 0, we have all the zeros of the function. Understanding complex roots is essential because it tells us that the graph of the function does not intersect the x-axis at these points. Complex roots often arise in polynomial equations of degree 2 or higher and are crucial in fields such as electrical engineering and quantum mechanics.

So, the zeros of the function h(x) = 3x³ + 9x² + 12x are:

1. x = 0 (a real zero)
2. x = (-3 + i√7) / 2 (a complex zero)
3. x = (-3 - i√7) / 2 (a complex zero)

Conclusion

In summary, we have successfully found all the zeros of the function h(x) = 3x³ + 9x² + 12x. We started by factoring out the greatest common factor, which simplified the equation. Then, we used the zero-product property to set each factor equal to zero. We solved the resulting linear equation and used the quadratic formula to solve the quadratic equation. This process highlights the importance of various algebraic techniques and the usefulness of complex numbers in finding all the solutions to polynomial equations. Understanding how to find zeros is a fundamental skill in mathematics with applications in various fields.

Remember, finding the zeros of a function is a crucial skill in mathematics. It involves understanding different algebraic techniques and applying them systematically. By following these steps, you can confidently find the zeros of similar cubic functions. Keep practicing, and you'll master this essential mathematical skill! For further learning and resources, you can visit trusted websites like Khan Academy.