Factorization of a Quadratic Expression
In this article, we will explore the factorization of the quadratic expression:
4x² + 9y² + 16z² + 12xy - 24yz - 16xz
Prerequisites
Before diving into the factorization, it's essential to have a basic understanding of quadratic expressions and their properties. A quadratic expression is a polynomial expression of degree two, which means the highest power of the variable(s) is two.
Factoring the Expression
To factor the given expression, we need to identify the greatest common factor (GCF) among the terms. After examining the expression, we can see that the GCF is 1. Therefore, we cannot factor out any common factor.
Next, we can try to group the terms to identify any patterns or relationships between them. Let's rearrange the terms to group the x-terms, y-terms, and z-terms:
(4x² - 16xz) + (9y² - 24yz) + (16z²)
Now, we can try to factor each group separately:
x-terms
4x² - 16xz = 4x(x - 4z)
y-terms
9y² - 24yz = 3y(3y - 8z)
z-terms
16z² (this term cannot be factored further)
Now that we have factored each group, we can combine them to get the final factored form:
Factored Form
4x² + 9y² + 16z² + 12xy - 24yz - 16xz = 4x(x - 4z) + 3y(3y - 8z) + 16z²
Conclusion
In this article, we have successfully factored the quadratic expression 4x² + 9y² + 16z² + 12xy - 24yz - 16xz. The factored form reveals the underlying structure of the expression, which can be useful in various mathematical applications.
