What Is The Factored Form Of $2x^3 + 4x^2 - X$?A. $2x(x^2 + 2x + 1)$B. \$x(2x^2 + 4x + 1)$[/tex\]C. $2x(x^2 + 2x - 1)$D. $x(2x^2 + 4x - 1)$

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Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will explore the factored form of the polynomial $2x^3 + 4x^2 - x$ and examine the different options provided.

Understanding the Polynomial

Before we dive into factoring, let's take a closer look at the given polynomial: $2x^3 + 4x^2 - x$. This polynomial is a cubic polynomial, meaning it has a degree of 3. The leading coefficient is 2, and the variable is x.

Factoring by Grouping

One common method for factoring polynomials is factoring by grouping. This involves grouping the terms of the polynomial into pairs and then factoring out the greatest common factor (GCF) from each pair.

Let's apply this method to the given polynomial:

2x3+4x2−x2x^3 + 4x^2 - x

We can group the first two terms together:

2x3+4x22x^3 + 4x^2

The GCF of these two terms is 2x^2. Factoring out 2x^2 from the first two terms, we get:

2x2(x+2)2x^2(x + 2)

Now, let's look at the third term:

−x-x

We can factor out -1 from the third term:

−x-x

Now, we have:

2x2(x+2)−x2x^2(x + 2) - x

We can factor out x from both terms:

x(2x2+4x+1)x(2x^2 + 4x + 1)

Comparing with the Options

Now that we have factored the polynomial, let's compare our result with the options provided:

A. $2x(x^2 + 2x + 1)$

B. $x(2x^2 + 4x + 1)$

C. $2x(x^2 + 2x - 1)$

D. $x(2x^2 + 4x - 1)$

Our factored form is:

x(2x2+4x+1)x(2x^2 + 4x + 1)

This matches option B.

Conclusion

In this article, we explored the factored form of the polynomial $2x^3 + 4x^2 - x$. We applied the method of factoring by grouping and obtained the factored form $x(2x^2 + 4x + 1)$. This matches option B. Factoring polynomials is an essential skill in algebra, and by following the steps outlined in this article, you can master this technique and apply it to a wide range of problems.

Additional Tips and Tricks

  • When factoring by grouping, make sure to identify the GCF of each pair of terms.
  • When factoring out a common factor, make sure to include all the terms that have that factor.
  • When comparing your result with the options provided, make sure to check the coefficients and the variables carefully.

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In our previous article, we explored the factored form of the polynomial $2x^3 + 4x^2 - x$ and examined the different options provided. In this article, we will answer some frequently asked questions about factoring polynomials.

Q&A

Q: What is factoring by grouping?

A: Factoring by grouping is a method of factoring polynomials that involves grouping the terms of the polynomial into pairs and then factoring out the greatest common factor (GCF) from each pair.

Q: How do I identify the GCF of a pair of terms?

A: To identify the GCF of a pair of terms, look for the largest factor that divides both terms. You can use the distributive property to factor out the GCF from each term.

Q: What is the difference between factoring by grouping and factoring out a common factor?

A: Factoring by grouping involves factoring out the GCF from each pair of terms, while factoring out a common factor involves factoring out a common factor from all the terms.

Q: Can I factor a polynomial that has no common factors?

A: Yes, you can factor a polynomial that has no common factors by using the method of factoring by grouping or by using other factoring techniques such as factoring by difference of squares or factoring by sum and difference.

Q: How do I know which factoring technique to use?

A: The choice of factoring technique depends on the type of polynomial you are working with. For example, if you are working with a polynomial that has a difference of squares, you can use the factoring by difference of squares technique.

Q: Can I factor a polynomial that has a variable in the denominator?

A: No, you cannot factor a polynomial that has a variable in the denominator. You can simplify the polynomial by canceling out any common factors, but you cannot factor it.

Q: How do I check my work when factoring a polynomial?

A: To check your work, multiply the factors together to see if you get the original polynomial. You can also use the distributive property to expand the factors and see if you get the original polynomial.

Common Mistakes to Avoid

  • Not identifying the GCF of a pair of terms
  • Factoring out a common factor from only one term
  • Not checking your work when factoring a polynomial
  • Using the wrong factoring technique for the type of polynomial you are working with

Conclusion

In this article, we answered some frequently asked questions about factoring polynomials. By following the tips and tricks outlined in this article, you can become proficient in factoring polynomials and tackle even the most challenging problems with confidence.

Additional Resources

  • For more information on factoring polynomials, check out our previous article on the factored form of the polynomial $2x^3 + 4x^2 - x$.
  • For practice problems and exercises, try using online resources such as Khan Academy or Mathway.
  • For more advanced topics in algebra, check out our article on solving systems of equations.

By following these resources and practicing regularly, you can master the art of factoring polynomials and become a proficient algebraist.