Factored Form Of 2x³ + 4x² - X Step-by-Step Solution

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Factoring polynomials is a fundamental skill in algebra, and it's crucial for solving equations, simplifying expressions, and understanding the behavior of functions. In this comprehensive guide, we'll break down the process of factoring the polynomial 2x³ + 4x² - x, providing a step-by-step explanation that's easy to follow. We'll explore the concept of factoring, common factoring techniques, and apply these techniques to our specific polynomial. By the end of this article, you'll not only know the factored form of 2x³ + 4x² - x but also understand the underlying principles that make factoring possible.

Understanding Factoring

At its core, factoring is the reverse process of expanding. When we expand, we multiply expressions together to get a single polynomial. Factoring, on the other hand, involves breaking down a polynomial into a product of simpler expressions. These simpler expressions are called factors. For instance, consider the number 12. It can be factored as 2 × 6 or 3 × 4, where 2, 6, 3, and 4 are factors of 12. Similarly, polynomials can be factored into simpler polynomial expressions.

Factoring polynomials is essential for various algebraic manipulations. It allows us to simplify complex expressions, solve polynomial equations, and analyze the roots and zeros of polynomial functions. Mastery of factoring techniques is a cornerstone of algebraic proficiency.

Key Factoring Techniques

Several techniques can be employed to factor polynomials, each suited for different types of expressions. Some of the most common techniques include:

  • Greatest Common Factor (GCF): This involves identifying the largest factor common to all terms in the polynomial and factoring it out.
  • Difference of Squares: This applies to binomials in the form a² - b², which can be factored as (a + b)(a - b).
  • Perfect Square Trinomials: These are trinomials in the form a² + 2ab + b² or a² - 2ab + b², which can be factored as (a + b)² or (a - b)², respectively.
  • Factoring by Grouping: This technique is used for polynomials with four or more terms and involves grouping terms with common factors.
  • Trial and Error: This method is often used for quadratic trinomials and involves systematically testing different combinations of factors.

In the case of our polynomial, 2x³ + 4x² - x, we'll primarily focus on the Greatest Common Factor (GCF) technique, as it's the most direct approach for this particular expression.

Step-by-Step Factoring of 2x³ + 4x² - x

Let's now apply the factoring techniques to our polynomial, 2x³ + 4x² - x. Here's a step-by-step breakdown:

1. Identify the Greatest Common Factor (GCF)

The first step in factoring any polynomial is to look for the Greatest Common Factor (GCF) of all the terms. The GCF is the largest factor that divides evenly into each term. In our polynomial, 2x³ + 4x² - x, we have three terms: 2x³, 4x², and -x.

Let's analyze the coefficients and the variables separately:

  • Coefficients: The coefficients are 2, 4, and -1. The greatest common factor of these numbers is 1, as 1 is the largest number that divides evenly into all three.
  • Variables: The variables are x³, x², and x. The greatest common factor of these terms is x, as it's the lowest power of x present in all terms.

Therefore, the Greatest Common Factor (GCF) of the entire polynomial 2x³ + 4x² - x is x.

2. Factor out the GCF

Now that we've identified the GCF as x, we can factor it out from each term in the polynomial. This involves dividing each term by the GCF and writing the GCF outside the parentheses.

Dividing each term by x, we get:

  • 2x³ / x = 2x²
  • 4x² / x = 4x
  • -x / x = -1

Now, we can rewrite the polynomial as the GCF multiplied by the result of the division:

2x³ + 4x² - x = x(2x² + 4x - 1)

3. Check for Further Factoring

After factoring out the GCF, it's essential to check if the remaining expression inside the parentheses can be factored further. In our case, the expression inside the parentheses is 2x² + 4x - 1. This is a quadratic trinomial.

To determine if this quadratic trinomial can be factored further, we can attempt to factor it using techniques like trial and error or by checking its discriminant. However, in this particular case, the quadratic trinomial 2x² + 4x - 1 does not factor nicely into integer factors. Its discriminant (b² - 4ac) is 4² - 4(2)(-1) = 16 + 8 = 24, which is not a perfect square, indicating that the roots are irrational and the trinomial cannot be factored further using simple integer coefficients.

4. Final Factored Form

Since the expression inside the parentheses, 2x² + 4x - 1, cannot be factored further, we have reached the final factored form of the polynomial:

2x³ + 4x² - x = x(2x² + 4x - 1)

This is the completely factored form of the given polynomial.

Conclusion

In this guide, we've successfully factored the polynomial 2x³ + 4x² - x. We began by understanding the concept of factoring and its importance in algebra. We then identified the Greatest Common Factor (GCF) of the terms, which was x. By factoring out the GCF, we arrived at the factored form: x(2x² + 4x - 1). We also verified that the quadratic trinomial 2x² + 4x - 1 could not be factored further using simple integer coefficients.

Factoring polynomials is a vital skill in algebra, and by mastering these techniques, you'll be well-equipped to tackle more complex algebraic problems. Remember to always look for the GCF first, and then explore other factoring methods if necessary. With practice, you'll become proficient at factoring various types of polynomials.

Therefore, the correct factored form of 2x³ + 4x² - x is x(2x² + 4x - 1). This corresponds to option C in the multiple-choice question.

Choosing the Correct Option

Based on our step-by-step factoring process, we've determined that the factored form of 2x³ + 4x² - x is x(2x² + 4x - 1). Let's now compare this result with the given options:

A. x(2x² + 4x + 1) B. 2x(x² + 2x + 1) C. x(2x² + 4x - 1) D. 2x(x² + 2x - 1)

By comparing our factored form with the options, we can clearly see that option C, x(2x² + 4x - 1), matches our result. Therefore, option C is the correct answer.

Practice Problems

To solidify your understanding of factoring, try factoring the following polynomials:

  1. 3x³ - 6x² + 9x
  2. 4x² - 16
  3. x² + 5x + 6

Factoring is a skill that improves with practice, so the more you work with different types of polynomials, the more confident you'll become in your ability to factor them. Remember to always look for the GCF first and then consider other factoring techniques as needed.

By working through these practice problems, you'll further develop your factoring skills and gain a deeper understanding of this important algebraic concept.