Solving for the Zeros of a Cubic Polynomial: A Step-by-Step Guide

In mathematics, polynomials are a fundamental part of algebra, and cubic polynomials are particularly intriguing due to their complex structures and intriguing zeros. In this article, we will explore how to find the zeros of a given cubic polynomial by following the steps of factorization.

## 1. Understanding the Problem Consider the cubic polynomial equation:

x^3 – 3x^2 – 124

The problem statement suggests that '4' is one of the zeros of this polynomial. This means that when the polynomial is evaluated at x4, the result is zero. To solve for the other two zeros, we need to factorize the polynomial or use synthetic division and other algebraic techniques.

## 2. Factorization Approach To factorize the given polynomial, we start by assuming that one of its zeros is 4. This allows us to rewrite the polynomial in the form:

(x - 4)(ax^2 bx c)

This form is derived from the fact that if 4 is a root, then (x - 4) is a factor of the polynomial. Now, let's follow the steps to find the values of a, b, and c.

## 3. Detailed Steps to Solve

First, let's restate the original equation:

x^3 – 3x^2 – 124 0

To begin the factorization, we can use synthetic division. This method is efficient and simplifies the process of finding the factors. The synthetic division setup looks like this:

Performing the synthetic division, we get the following:

1 -3 -1 -24 4 16 -32 -200 --------- 1 -2 -1 -4

From the synthetic division, we can see that the coefficients of the quadratic polynomial are 1, -2, -4. Thus, our polynomial can be written as:

x^3 – 3x^2 – 124 (x - 4)(x^2 - 2x - 32)

To find the other zeros, we need to solve the quadratic equation:

x^2 - 2x - 32 0

## 4. Solving the Quadratic Equation To solve the quadratic equation x^2 - 2x - 32 0, we can use the quadratic formula:

x [-b ± sqrt(b^2 - 4ac)] / (2a)

Here, a 1, b -2, c -32. Plugging these values into the formula:

x [2 ± sqrt((-2)^2 - 4 * 1 * -32)] / (2 * 1)

x [2 ± sqrt(4 128)] / 2

x [2 ± sqrt(132)] / 2

Further simplifying, we get:

x [2 ± sqrt(4 * 33)] / 2

x [2 ± 2 * sqrt(33)] / 2

x 1 ± sqrt(33)

Therefore, the zeros of the quadratic equation are:

x 1 sqrt(33)

x 1 - sqrt(33)

## 5. Conclusion In conclusion, we have found that the zeros of the cubic polynomial x^3 - 3x^2 - 124 are 4, 1 sqrt(33), and 1 - sqrt(33). The factorization process is a powerful method to find zeros of a polynomial, and it simplifies complex equations into manageable parts.

Understanding the steps of factorization, solving quadratic equations, and applying synthetic division can greatly enhance one's problem-solving skills in algebra. These techniques are not only useful in academic settings but also have practical applications in various fields, such as engineering, physics, and economics.