what is the completely factored form of 3x5-7x4+6x2-14x

2 min read 21-02-2025

what is the completely factored form of 3x5-7x4+6x2-14x

Factoring polynomials is a crucial skill in algebra. It involves breaking down a complex expression into simpler components, often revealing underlying relationships and solutions. This article will guide you through the process of completely factoring the polynomial 3x⁵ - 7x⁴ + 6x² - 14x.

Understanding Polynomial Factoring

Before diving into the specific problem, let's refresh the basics of polynomial factoring. The goal is to express the polynomial as a product of simpler polynomials. Common techniques include:

Greatest Common Factor (GCF): Identifying the largest term that divides all terms of the polynomial.
Grouping: Rearranging terms and factoring in groups to reveal common factors.
Special Factoring Patterns: Recognizing patterns like difference of squares (a² - b²) or perfect squares (a² + 2ab + b²).

Factoring 3x⁵ - 7x⁴ + 6x² - 14x Step-by-Step

Let's apply these techniques to factor 3x⁵ - 7x⁴ + 6x² - 14x:

Step 1: Find the Greatest Common Factor (GCF)

The first step in any factoring problem is to look for a GCF. Notice that each term in the polynomial contains an 'x'. We can factor out an 'x' :

x(3x⁴ - 7x³ + 6x - 14)

Step 2: Grouping

Now, we have a four-term polynomial inside the parentheses. Let's try grouping. We'll group the first two terms and the last two terms:

x[(3x⁴ - 7x³) + (6x - 14)]

Step 3: Factor Each Group

Now factor out the GCF from each group:

x[x³(3x - 7) + 2(3x - 7)]

Step 4: Identify and Factor the Common Binomial

Notice that both terms inside the brackets now share a common binomial factor (3x - 7). We can factor that out:

x(3x - 7)(x³ + 2)

Step 5: Check for Further Factoring

At this point, we need to determine if (x³ + 2) can be factored further. While x³ + 2 is a sum of cubes, the sum of cubes factoring formula only applies when the terms are perfect cubes. In this case, 2 is not a perfect cube, so it cannot be factored further using standard techniques.

The Completely Factored Form

Therefore, the completely factored form of the polynomial 3x⁵ - 7x⁴ + 6x² - 14x is:

x(3x - 7)(x³ + 2)

Conclusion

By systematically applying the GCF and grouping methods, we've successfully factored the given polynomial. Remember to always check for further factoring opportunities after each step. This example demonstrates a common approach to factoring polynomials, combining multiple techniques for a complete solution. The final factored form, x(3x - 7)(x³ + 2), represents the polynomial in its simplest and most useful form for many algebraic manipulations.