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

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.

What is Factoring?

Factoring is the process of expressing a polynomial as a product of simpler polynomials, called factors. Each factor is a polynomial that, when multiplied together, results in the original polynomial. Factoring is an essential tool in algebra, as it allows us to simplify complex expressions, solve equations, and identify the roots of a polynomial.

The Factored Form of a Polynomial

The factored form of a polynomial is a product of one or more factors, each of which is a polynomial. The general form of a factored polynomial is:

$$a(x - r_1)(x - r_2)...(x - r_n)$$

where $a$ is a constant, and $r_1, r_2,...,r_n$ are the roots of the polynomial.

Factoring the Polynomial $2x^3 + 4x^2 - x$

To factor the polynomial $2x^3 + 4x^2 - x$, we need to identify the greatest common factor (GCF) of the terms. The GCF is the largest polynomial that divides each term of the polynomial.

Step 1: Identify the GCF

The GCF of the terms $2x^3$, $4x^2$, and $-x$ is $x$. Therefore, we can factor out an $x$ from each term:

$$2x^3 + 4x^2 - x = x(2x^2 + 4x - 1)$$

Step 2: Factor the Quadratic Expression

The quadratic expression $2x^2 + 4x - 1$ can be factored using the quadratic formula or by finding two numbers whose product is $-2$ and whose sum is $4$. The numbers are $2$ and $-1$, so we can write:

$$2x^2 + 4x - 1 = (2x + 1)(x + 1)$$

Step 3: Write the Factored Form

Now that we have factored the quadratic expression, we can write the factored form of the polynomial:

$$2x^3 + 4x^2 - x = x(2x + 1)(x + 1)$$

Comparing the Factored Form with the Options

Now that we have found the factored form of the polynomial, we can compare it with the options provided:

A. $2x(x^2 + 2x + 1)$ B. $x(2x^2 + 4x + 1)$ C. $2(2, 2)$ D. $x(2x^2 + 4x - 1)$

The only option that matches our factored form is:

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

Conclusion

In this article, we have explored the factored form of the polynomial $2x^3 + 4x^2 - x$ and compared it with the options provided. We have shown that the factored form of the polynomial is $x(2x^2 + 4x - 1)$, which matches option D. Factoring polynomials is an essential tool in algebra, and we hope that this article has provided a clear and concise guide to factoring polynomials.

Final Answer

The final answer is:

Introduction

In our previous article, we explored the factored form of the polynomial $2x^3 + 4x^2 - x$ and compared it with the options provided. In this article, we will answer some frequently asked questions about factoring polynomials.

Q&A

Q: What is the greatest common factor (GCF) of a polynomial?

A: The GCF of a polynomial is the largest polynomial that divides each term of the polynomial.

Q: How do I identify the GCF of a polynomial?

A: To identify the GCF of a polynomial, you need to find the largest polynomial that divides each term of the polynomial. You can do this by looking for the largest polynomial that divides each term.

Q: What is the difference between factoring and simplifying a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials, while simplifying a polynomial involves combining like terms to reduce the polynomial to its simplest form.

Q: Can I factor a polynomial with a negative coefficient?

A: Yes, you can factor a polynomial with a negative coefficient. To do this, you need to multiply each term of the polynomial by -1.

Q: How do I factor a polynomial with a variable in the denominator?

A: To factor a polynomial with a variable in the denominator, you need to multiply each term of the polynomial by the reciprocal of the variable.

Q: Can I factor a polynomial with a fractional coefficient?

A: Yes, you can factor a polynomial with a fractional coefficient. To do this, you need to multiply each term of the polynomial by the reciprocal of the fraction.

Q: What is the difference between factoring and solving a polynomial equation?

A: Factoring a polynomial equation involves expressing it as a product of simpler polynomials, while solving a polynomial equation involves finding the values of the variable that satisfy the equation.

Q: Can I factor a polynomial with a complex coefficient?

A: Yes, you can factor a polynomial with a complex coefficient. To do this, you need to multiply each term of the polynomial by the conjugate of the complex coefficient.

Common Mistakes to Avoid

When factoring polynomials, there are several common mistakes to avoid:

• Not identifying the GCF: Failing to identify the GCF of a polynomial can lead to incorrect factoring.
• Not combining like terms: Failing to combine like terms can lead to incorrect simplification.
• Not multiplying each term by the reciprocal: Failing to multiply each term by the reciprocal of a variable in the denominator can lead to incorrect factoring.
• Not multiplying each term by the conjugate: Failing to multiply each term by the conjugate of a complex coefficient can lead to incorrect factoring.

Conclusion

In this article, we have answered some frequently asked questions about factoring polynomials. We have also highlighted some common mistakes to avoid when factoring polynomials. By following these tips and avoiding these common mistakes, you can become proficient in factoring polynomials and solve a wide range of problems.

Final Tips

• Practice, practice, practice: The more you practice factoring polynomials, the more comfortable you will become with the process.
• Use online resources: There are many online resources available that can help you learn how to factor polynomials, including video tutorials and practice problems.
• Seek help when needed: If you are struggling with a particular problem or concept, don't be afraid to seek help from a teacher or tutor.

Final Answer

The final answer is: There is no final answer, as this article is a Q&A guide. However, we hope that you have found the information helpful and that you will continue to practice and improve your skills in factoring polynomials.