Factor The Polynomial Completely.Find A GCF: − 2 X 2 + 2 + 5 X 3 − 5 X -2x^2 + 2 + 5x^3 - 5x − 2 X 2 + 2 + 5 X 3 − 5 X GCF: − 2 -2 − 2 , GCF: 5 X 5x 5 X Factor Out The GCF: − 2 ( X 2 − 1 ) + 5 X ( X 2 − 1 -2(x^2 - 1) + 5x(x^2 - 1 − 2 ( X 2 − 1 ) + 5 X ( X 2 − 1 ]Which Product Of Prime Polynomials Is Equivalent To The Original Polynomial?A.

Introduction

Factoring polynomials is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. In this article, we will focus on factoring the polynomial completely, finding the greatest common factor (GCF), and identifying the product of prime polynomials equivalent to the original polynomial. We will use the given polynomial $-2x^2 + 2 + 5x^3 - 5x$ as an example to demonstrate the step-by-step process.

Find a GCF

To factor the polynomial completely, we need to find the greatest common factor (GCF) of the terms. The GCF is the largest expression that divides each term of the polynomial without leaving a remainder.

• GCF: $-2$

The first step is to identify the common factors among the terms. In this case, we can see that the terms $-2x^2$ and $-5x$ have a common factor of $-2$. Therefore, the GCF is $-2$.

• GCF: $5x$

Similarly, we can identify another common factor among the terms. The terms $5x^3$ and $-5x$ have a common factor of $5x$. Therefore, the GCF is $5x$.

Factor out the GCF

Now that we have identified the GCFs, we can factor them out of the polynomial.

• Factor out the GCF: $-2$

To factor out the GCF $-2$, we divide each term by $-2$.

$$\frac{-2x^2}{-2} + \frac{2}{-2} + \frac{5x^3}{-2} - \frac{5x}{-2}$$

This simplifies to:

$$x^2 - 1 + \frac{5x^3}{-2} - \frac{5x}{2}$$

• Factor out the GCF: $5x$

To factor out the GCF $5x$, we divide each term by $5x$.

$$\frac{5x^3}{5x} - \frac{5x}{5x} - \frac{2x^2}{5x} + \frac{2}{5x}$$

This simplifies to:

$$x^2 - \frac{1}{5} + x - \frac{2}{5}$$

Combine the GCFs

Now that we have factored out the GCFs, we can combine them to simplify the polynomial.

$$-2(x^2 - 1) + 5x(x^2 - 1)$$

This simplifies to:

$$(5x - 2)(x^2 - 1)$$

Product of Prime Polynomials

A prime polynomial is a polynomial that cannot be factored further into the product of two or more polynomials. In this case, we have factored the polynomial completely, and we can see that the product of prime polynomials equivalent to the original polynomial is:

$$(5x - 2)(x^2 - 1)$$

This is the final answer.

Conclusion

Factoring polynomials is a crucial concept in algebra that helps us simplify complex expressions and solve equations. By finding the greatest common factor (GCF) and factoring it out, we can simplify the polynomial and identify the product of prime polynomials equivalent to the original polynomial. In this article, we used the given polynomial $-2x^2 + 2 + 5x^3 - 5x$ as an example to demonstrate the step-by-step process. We identified the GCFs, factored them out, and combined them to simplify the polynomial. The final answer is the product of prime polynomials equivalent to the original polynomial.

References

• [1] Algebra, 2nd ed. by Michael Artin. Prentice Hall, 2010.
• [2] Calculus, 3rd ed. by Michael Spivak. Publish or Perish, 2008.

Additional Resources

• Khan Academy: Factoring Polynomials
• Mathway: Factoring Polynomials
• Wolfram Alpha: Factoring Polynomials
  Q&A: Factoring Polynomials Completely =============================================

Introduction

Factoring polynomials is a fundamental concept in algebra that helps us simplify complex expressions and solve equations. In our previous article, we discussed how to factor the polynomial completely, find the greatest common factor (GCF), and identify the product of prime polynomials equivalent to the original polynomial. In this article, we will answer some frequently asked questions (FAQs) related to factoring polynomials completely.

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

A: The greatest common factor (GCF) of a polynomial is the largest expression that divides each term of the polynomial without leaving a remainder.

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

A: To find the GCF of a polynomial, you need to identify the common factors among the terms. You can do this by looking for the largest expression that divides each term without leaving a remainder.

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 its complexity.

Q: Can a polynomial have more than one GCF?

A: Yes, a polynomial can have more than one GCF. In fact, it's common for a polynomial to have multiple GCFs, especially when it has multiple terms with common factors.

Q: How do I factor out the GCF of a polynomial?

A: To factor out the GCF of a polynomial, you need to divide each term by the GCF. This will give you a new polynomial that is simpler than the original one.

Q: What is the product of prime polynomials?

A: The product of prime polynomials is a polynomial that cannot be factored further into the product of two or more polynomials.

Q: How do I identify the product of prime polynomials?

A: To identify the product of prime polynomials, you need to factor the polynomial completely and look for the simplest polynomials that cannot be factored further.

Q: Can a polynomial have multiple products of prime polynomials?

A: Yes, a polynomial can have multiple products of prime polynomials. In fact, it's common for a polynomial to have multiple products of prime polynomials, especially when it has multiple terms with common factors.

Q: How do I simplify a polynomial using factoring?

A: To simplify a polynomial using factoring, you need to factor the polynomial completely and then combine like terms to reduce its complexity.

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 correctly
• Not factoring out the GCF correctly
• Not combining like terms correctly
• Not checking for multiple GCFs

Conclusion

Factoring polynomials is a crucial concept in algebra that helps us simplify complex expressions and solve equations. By understanding the GCF, factoring, and simplifying polynomials, we can solve a wide range of problems in mathematics and other fields. In this article, we answered some frequently asked questions (FAQs) related to factoring polynomials completely. We hope this article has been helpful in clarifying any doubts you may have had about factoring polynomials.

References

• [1] Algebra, 2nd ed. by Michael Artin. Prentice Hall, 2010.
• [2] Calculus, 3rd ed. by Michael Spivak. Publish or Perish, 2008.

Additional Resources

• Khan Academy: Factoring Polynomials
• Mathway: Factoring Polynomials
• Wolfram Alpha: Factoring Polynomials