Select The Correct Answer.What Is The Completely Factored Form Of This Polynomial? 2 X 4 + 11 X 2 + 5 2x^4 + 11x^2 + 5 2 X 4 + 11 X 2 + 5 A. ( 2 X 2 + 1 ) ( X 2 + 5 (2x^2 + 1)(x^2 + 5 ( 2 X 2 + 1 ) ( X 2 + 5 ] B. ( 2 X + 1 ) ( X + 5 (2x + 1)(x + 5 ( 2 X + 1 ) ( X + 5 ] C. ( 2 X + 5 ) ( X + 1 (2x + 5)(x + 1 ( 2 X + 5 ) ( X + 1 ] D. ( 2 X 2 + 5 ) ( X 2 + 1 (2x^2 + 5)(x^2 + 1 ( 2 X 2 + 5 ) ( X 2 + 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 process of factoring polynomials and apply it to the given polynomial $2x^4 + 11x^2 + 5$. We will examine each option and determine the correct answer.

What is Factoring?

Factoring a polynomial involves expressing it as a product of simpler polynomials, called factors. These factors can be linear or quadratic expressions. The process of factoring involves finding the common factors of the polynomial and expressing it as a product of these factors.

Types of Factoring

There are several types of factoring, including:

• Greatest Common Factor (GCF): This involves finding the greatest common factor of the polynomial and expressing it as a product of the GCF and the remaining polynomial.
• Difference of Squares: This involves expressing a polynomial as the difference of two squares.
• Sum and Difference: This involves expressing a polynomial as the sum or difference of two or more polynomials.
• Quadratic Formula: This involves expressing a quadratic polynomial in the form of $ax^2 + bx + c = 0$.

Factoring the Given Polynomial

The given polynomial is $2x^4 + 11x^2 + 5$. To factor this polynomial, we need to find the common factors and express it as a product of simpler polynomials.

Option A: $(2x^2 + 1)(x^2 + 5)$

Let's examine option A: $(2x^2 + 1)(x^2 + 5)$. To verify this option, we need to multiply the two factors and check if it equals the original polynomial.

$(2x^2 + 1)(x^2 + 5) = 2x^4 + 10x^2 + x^2 + 5$

Simplifying the expression, we get:

$2x^4 + 11x^2 + 5$

This matches the original polynomial, so option A is a possible solution.

Option B: $(2x + 1)(x + 5)$

Let's examine option B: $(2x + 1)(x + 5)$. To verify this option, we need to multiply the two factors and check if it equals the original polynomial.

$(2x + 1)(x + 5) = 2x^2 + 10x + x + 5$

Simplifying the expression, we get:

$2x^2 + 11x + 5$

This does not match the original polynomial, so option B is not a possible solution.

Option C: $(2x + 5)(x + 1)$

Let's examine option C: $(2x + 5)(x + 1)$. To verify this option, we need to multiply the two factors and check if it equals the original polynomial.

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

Simplifying the expression, we get:

$2x^2 + 7x + 5$

This does not match the original polynomial, so option C is not a possible solution.

Option D: $(2x^2 + 5)(x^2 + 1)$

Let's examine option D: $(2x^2 + 5)(x^2 + 1)$. To verify this option, we need to multiply the two factors and check if it equals the original polynomial.

$(2x^2 + 5)(x^2 + 1) = 2x^4 + 2x^2 + 5x^2 + 5$

Simplifying the expression, we get:

$2x^4 + 7x^2 + 5$

This does not match the original polynomial, so option D is not a possible solution.

Conclusion

Based on our analysis, we can conclude that the correct answer is option A: $(2x^2 + 1)(x^2 + 5)$. This option matches the original polynomial, and the other options do not.

Final Answer

The final answer is: A. $(2x^2 + 1)(x^2 + 5)$

Additional Tips and Resources

• To factor a polynomial, start by finding the greatest common factor (GCF) of the polynomial.
• Use the difference of squares formula to factor polynomials in the form of $a^2 - b^2$.
• Use the sum and difference formula to factor polynomials in the form of $a^2 + b^2$.
• Use the quadratic formula to factor quadratic polynomials in the form of $ax^2 + bx + c = 0$.

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

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 provide a Q&A guide to help you understand the process of factoring polynomials and address common questions and concerns.

Q: What is factoring?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials, called factors. These factors can be linear or quadratic expressions.

Q: Why is factoring important?

A: Factoring is important because it allows us to simplify complex polynomials and make them easier to work with. It also helps us to identify the roots of a polynomial, which is essential in many areas of mathematics and science.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

• Greatest Common Factor (GCF): This involves finding the greatest common factor of the polynomial and expressing it as a product of the GCF and the remaining polynomial.
• Difference of Squares: This involves expressing a polynomial as the difference of two squares.
• Sum and Difference: This involves expressing a polynomial as the sum or difference of two or more polynomials.
• Quadratic Formula: This involves expressing a quadratic polynomial in the form of $ax^2 + bx + c = 0$.

Q: How do I factor a polynomial?

A: To factor a polynomial, start by finding the greatest common factor (GCF) of the polynomial. Then, use the difference of squares formula to factor polynomials in the form of $a^2 - b^2$. Use the sum and difference formula to factor polynomials in the form of $a^2 + b^2$. Finally, use the quadratic formula to factor quadratic polynomials in the form of $ax^2 + bx + c = 0$.

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

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

• Not finding the greatest common factor (GCF): Make sure to find the GCF of the polynomial before attempting to factor it.
• Not using the correct formula: Use the correct formula for the type of factoring you are doing.
• Not checking your work: Make sure to check your work by multiplying the factors together to ensure that they equal the original polynomial.

Q: How do I know if a polynomial can be factored?

A: A polynomial can be factored if it can be expressed as a product of simpler polynomials. To determine if a polynomial can be factored, try to find the greatest common factor (GCF) of the polynomial. If the GCF is a polynomial of degree less than the original polynomial, then the polynomial can be factored.

Q: What are some real-world applications of factoring polynomials?

A: Factoring polynomials has many real-world applications, including:

• Engineering: Factoring polynomials is used in engineering to design and analyze complex systems.
• Computer Science: Factoring polynomials is used in computer science to develop algorithms and solve problems.
• Economics: Factoring polynomials is used in economics to model and analyze economic systems.

Conclusion

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. By understanding the different types of factoring and how to factor polynomials, you can simplify complex polynomials and make them easier to work with. Remember to avoid common mistakes and check your work to ensure that you are factoring polynomials correctly.

Additional Resources

• Algebra textbooks: Check out algebra textbooks for more information on factoring polynomials.
• Online resources: Visit online resources such as Khan Academy, Mathway, and Wolfram Alpha for more information on factoring polynomials.
• Practice problems: Practice factoring polynomials with online resources or worksheets to improve your skills.

References

• "Algebra" by Michael Artin
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer

Note: The references provided are for general information and are not specific to the topic of factoring polynomials.