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

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 provide a step-by-step guide on how to factor it.

What is Factoring?

Factoring is the process of expressing a polynomial as a product of simpler polynomials. It involves finding the factors of the polynomial, which are the polynomials that multiply together to give the original polynomial. Factoring is an essential tool in algebra, as it allows us to simplify complex polynomials and solve equations.

The Factored Form of a Polynomial

The factored form of a polynomial is a product of simpler polynomials, each of which is a factor of the original polynomial. The factored form of a polynomial can be written in the form:

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

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 find the factors of the polynomial. We can start by factoring out the greatest common factor (GCF) of the polynomial, which is $x$. Factoring out $x$ gives us:

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

Now, we need to factor the quadratic expression $2x^2 + 4x - 1$. We can use the quadratic formula to find the roots of the quadratic expression:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a = 2$, $b = 4$, and $c = -1$. Plugging in these values, we get:

$$x = \frac{-4 \pm \sqrt{4^2 - 4(2)(-1)}}{2(2)}$$

Simplifying, we get:

$$x = \frac{-4 \pm \sqrt{16 + 8}}{4}$$

$$x = \frac{-4 \pm \sqrt{24}}{4}$$

$$x = \frac{-4 \pm 2\sqrt{6}}{4}$$

$$x = \frac{-2 \pm \sqrt{6}}{2}$$

Now, we can write the factored form of the polynomial as:

$$x(2x - 1)(x + \sqrt{6})$$

However, this is not one of the answer choices. We need to go back and try a different approach.

Alternative Approach

Let's try factoring the polynomial by grouping. We can group the first two terms together and the last term by itself:

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

Now, we can factor out the GCF of the first two terms, which is $2x^2$:

$$2x^2(2x + 2) - x$$

Next, we can factor out the GCF of the two terms, which is $x$:

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

This is one of the answer choices!

Conclusion

In this article, we explored the factored form of the polynomial $2x^3 + 4x^2 - x$. We used two different approaches to factor the polynomial: factoring out the GCF and factoring by grouping. We found that the factored form of the polynomial is $x(2x^2 + 4x - 1)$, which is one of the answer choices.

Answer

The correct answer is:

• D. $x(2x^2 + 4x - 1)$

Final Thoughts

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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 our previous article, we explored the factored form of the polynomial $2x^3 + 4x^2 - x$ and provided a step-by-step guide on how to factor it. In this article, we will answer some frequently asked questions about factoring polynomials.

Q&A

Q: What is factoring in algebra?

A: Factoring is the process of expressing a polynomial as a product of simpler polynomials. It involves finding the factors of the polynomial, which are the polynomials that multiply together to give the original polynomial.

Q: Why is factoring important in algebra?

A: Factoring is an essential tool in algebra that allows us to simplify complex polynomials and solve equations. It helps us to identify the roots of a polynomial, which is crucial in solving equations.

Q: What are the different methods of factoring polynomials?

A: There are several methods of factoring polynomials, including:

• Factoring out the greatest common factor (GCF)
• Factoring by grouping
• Factoring using the quadratic formula
• Factoring using the difference of squares formula

Q: How do I factor a polynomial with a GCF?

A: To factor a polynomial with a GCF, you need to identify the GCF and factor it out. For example, if you have the polynomial $6x^2 + 12x + 18$, the GCF is $6$. You can factor it out as follows:

$$6x^2 + 12x + 18 = 6(x^2 + 2x + 3)$$

Q: How do I factor a polynomial by grouping?

A: To factor a polynomial by grouping, you need to group the terms of the polynomial into pairs. For example, if you have the polynomial $x^2 + 4x + 4x + 16$, you can group the terms as follows:

$$(x^2 + 4x) + (4x + 16)$$

Now, you can factor out the GCF of each pair:

$$x(x + 4) + 4(x + 4)$$

Q: How do I factor a polynomial using the quadratic formula?

A: To factor a polynomial using the quadratic formula, you need to identify the coefficients of the polynomial and plug them into the quadratic formula. For example, if you have the polynomial $x^2 + 5x + 6$, you can use the quadratic formula as follows:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

where $a = 1$, $b = 5$, and $c = 6$. Plugging in these values, you get:

$$x = \frac{-5 \pm \sqrt{5^2 - 4(1)(6)}}{2(1)}$$

Simplifying, you get:

$$x = \frac{-5 \pm \sqrt{25 - 24}}{2}$$

$$x = \frac{-5 \pm \sqrt{1}}{2}$$

$$x = \frac{-5 \pm 1}{2}$$

Q: How do I factor a polynomial using the difference of squares formula?

A: To factor a polynomial using the difference of squares formula, you need to identify the difference of squares in the polynomial. For example, if you have the polynomial $x^2 - 9$, you can use the difference of squares formula as follows:

$$x^2 - 9 = (x - 3)(x + 3)$$

Q: What are some common mistakes to avoid when factoring polynomials?

A: Some common mistakes to avoid when factoring polynomials include:

• Not identifying the GCF of the polynomial
• Not grouping the terms correctly
• Not using the quadratic formula correctly
• Not using the difference of squares formula correctly

Q: How do I check if my factored form is correct?

A: To check if your factored form is correct, you need to multiply the factors together and see if you get the original polynomial. For example, if you have the factored form $x(x + 4) + 4(x + 4)$, you can multiply the factors together as follows:

$$(x)(x + 4) + 4(x + 4)$$

$$x^2 + 4x + 4x + 16$$

$$x^2 + 8x + 16$$

This is the original polynomial, so your factored form is correct.

Conclusion

In this article, we answered some frequently asked questions about factoring polynomials. We covered the different methods of factoring polynomials, including factoring out the GCF, factoring by grouping, factoring using the quadratic formula, and factoring using the difference of squares formula. We also covered some common mistakes to avoid when factoring polynomials and how to check if your factored form is correct.