Problem 25 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Find all zeros of \(f(x)\) (b) Write the complete factored form of \(f(x)\) $$ f(x)=3 x^{3}+3 x $$

Step-by-Step Solution

Answer

The zeros of \(f(x)\) are 0, \(i\), and \(-i\). The factored form is \(f(x) = 3x(x - i)(x + i)\).

Step 1: Factor Out Common Factors

First, identify and factor out the greatest common factor in the polynomial: \(f(x) = 3x^3 + 3x\). The greatest common factor here is \(3x\). Thus, we can rewrite the polynomial as: \(f(x) = 3x(x^2 + 1)\).

Step 2: Solve for Zero Values

Find the values of \(x\) that make the equation equal to zero. Start with \(3x = 0\). Solve for \(x\) by dividing both sides by 3 to get \(x = 0\).

Step 3: Analyze the Quadratic Polynomial

Next, consider the quadratic polynomial \(x^2 + 1\). The equation \(x^2 + 1 = 0\) has no real solutions because \(x^2 = -1\) is not possible with real numbers. To find the complex solutions, set \(x^2 = -1\) and solve, giving \(x = i\) or \(x = -i\), where \(i\) is the imaginary unit \(i^2 = -1\).

Step 4: Write the Complete Solution

Combine the solutions. The zeros of \(f(x)\) are \(x = 0, i, -i\).

Step 5: Write the Complete Factored Form

The complete factored form of the polynomial is \(f(x) = 3x(x - i)(x + i)\), which incorporates the complex roots \(x = i\) and \(x = -i\).

Key Concepts

Complex Numbers

Complex numbers are crucial when dealing with polynomial equations that do not have real solutions. In solving the polynomial equation \(x^2 + 1 = 0\), we encounter such a scenario. This equation suggests that \(x^2 = -1\), which does not yield a real number because no real number squared is negative.
To address this limitation, mathematicians defined a new type of number called imaginary numbers. The unit of imaginary numbers is denoted as \(i\), where \(i^2 = -1\). Hence, to solve \(x^2 + 1 = 0\), we can find solutions in the form of complex numbers, specifically \(x = i\) and \(x = -i\).
In general, complex numbers are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. They are used extensively in engineering, physics, and many areas of advanced mathematics, allowing us to solve equations that are not solvable within the realm of real numbers.

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that simplifies expressions and solves equations. The process involves breaking down a polynomial into a product of simpler and more manageable polynomials or constants. In the expression given, \(f(x) = 3x^3 + 3x\), the first step is to look for a common factor among the terms.
By factoring out the greatest common factor, which is \(3x\) in this case, we rewrite the expression as \(f(x) = 3x(x^2 + 1)\). This breakup allows us to handle each part separately, potentially offering insights into solving the equation for zeros or additional transformations.
Factoring serves as a vital tool not only in solving equations but also in simplifying expressions for other operations such as integration or differentiation in calculus. It is the first step in many mathematical procedures and provides a strong foundation for further studies in algebra and beyond.

Zeros of a Function

Zeros of a function, also known as roots or solutions, are the values of \(x\) where the function equals zero. These are critical in every mathematical field because they reveal important properties of functions.
For the polynomial \(f(x) = 3x(x^2 + 1)\), we identify the zeros by setting the function equal to zero and solving for \(x\). First, set \(3x = 0\), giving \(x = 0\) as a zero. Next, solve \(x^2 + 1 = 0\), which results in the complex solutions \(x = i\) and \(x = -i\).
Understanding the zeros is important because they provide the solution set of the equation, helping in graphing, analyzing, and understanding the behavior of the polynomial. Knowing how to find and interpret these zeros equips students with the skills needed for more advanced mathematical problem-solving.