Introduction
Algebraic expressions and equations play a crucial role in mathematics, science, and various fields of study. One fundamental aspect of working with algebraic expressions is factoring, which involves breaking down an expression into its constituent factors. In this article, we’ll delve into the world of factoring and explore the completely factored form of the polynomial expression 3x^5 – 7x^4 + 6x^2 – 14x.
Factoring Basics
Factoring is the process of rewriting an expression as a product of simpler expressions or factors. It is akin to breaking down a complex puzzle into smaller, more manageable pieces. When factoring, we look for common factors, often in the form of binomials or trinomials, that can be multiplied together to recreate the original expression.
The Completely Factored Form
The completely factored form of an expression is one where no further factoring is possible, meaning it has been reduced to its simplest terms. In the case of a polynomial like 3x^5 – 7x^4 + 6x^2 – 14x, factoring involves finding the factors that can be extracted from each term and combining like terms.
Step 1: Factor out the Greatest Common Factor (GCF)
The first step in factoring any polynomial is to identify and factor out the greatest common factor (GCF) among the terms. In our polynomial, the GCF is x, as it can be factored out from each term:
x(3x^4 – 7x^3 + 6x – 14)
Step 2: Factor by Grouping
Next, we group the terms inside the parentheses to see if any further factoring can be done. In this case, we can factor by grouping:
x(3x^4 – 7x^3 + 6x – 14) = x(3x^4 – 7x^3) + x(6x – 14)
Now, let’s factor each group separately:
x(3x^3(x – 7) + 2(3x – 7))
Step 3: Factor Out the Common Binomial
At this stage, we notice that both groups share a common binomial factor, which is (x – 7). We can factor this out:
x(3x^3 + 2)(x – 7)
Now, we have fully factored the original polynomial expression into its completely factored form.
Conclusion
The completely factored form of the polynomial expression 3x^5 – 7x^4 + 6x^2 – 14x is x(3x^3 + 2)(x – 7). Factoring allows us to simplify complex expressions, making them easier to work with and helping us identify important characteristics such as roots and critical points. Understanding factoring is a fundamental skill in algebra and lays the foundation for more advanced mathematical concepts.
