For the polynomial below, -2 and 2 are zeros.

Express as a product of linear factors.
,

To express the polynomial as a product of linear factors, we can start by using the given zeros, -2 and 2. This means that and are factors. We perform polynomial long division or synthetic division to divide by the product .

Upon performing the division, we will find that

Next, we can factor using the quadratic formula or completing the square, leading to two additional linear factors. We find that the factorization is:

So, the final expression in terms of linear factors is:

Now, let’s dive into some backdrop and practical applications of polynomials!

Polynomials have roots in the very foundation of mathematics, from the ancient Greeks studying equation solutions to the modern realm of computer graphics! In fact, polynomials are crucial for rendering curves and shapes in 3D animations – talk about bringing art to life with math!

In real-life applications, polynomials are used in various fields such as physics for modeling motion, economics for predicting growth trends, and engineering for stress analysis. Understanding how to factor polynomials not only aids in solving equations but also optimizes these models, sharpening our ability to make predictions and calculations in a world driven by data!