Factor Out The Common Factor From The Expression $2x^4 + 5x^3 - 8x - 20$.A. $(x^3 + 4)(-2x - 5$\]B. $(x^3 \cdot 4)(2x + 5$\]C. $(x^3 + 4)(2x + 5$\]

Factor Out the Common Factor from the Expression $2x^4 + 5x^3 - 8x - 20$

Understanding the Problem

Factoring out the common factor from an algebraic expression is a crucial step in simplifying and solving equations. In this problem, we are given the expression $2x^4 + 5x^3 - 8x - 20$ and we need to factor out the common factor. This involves identifying the common factor and expressing the given expression as a product of the common factor and another expression.

Identifying the Common Factor

To factor out the common factor, we need to identify the common factor among the terms in the given expression. The common factor is the term that divides each term in the expression without leaving a remainder. In this case, we can see that the term $x^3$ is common to the first two terms, and the term $-2$ is common to the last two terms.

Factoring Out the Common Factor

Now that we have identified the common factor, we can factor it out from the given expression. We can start by factoring out the common factor $x^3$ from the first two terms:

$2x^4 + 5x^3 - 8x - 20 = x^3(2x + 5) - 8x - 20$

Next, we can factor out the common factor $-2$ from the last two terms:

$x^3(2x + 5) - 8x - 20 = x^3(2x + 5) - 2(4x + 10)$

Now, we can see that the expression can be factored as:

$x^3(2x + 5) - 2(4x + 10) = (x^3 + 4)(2x + 5)$

Conclusion

In this problem, we have factored out the common factor from the expression $2x^4 + 5x^3 - 8x - 20$. We identified the common factor as $x^3$ and $-2$, and then factored it out from the given expression. The final factored form of the expression is $(x^3 + 4)(2x + 5)$.

Answer

The correct answer is:

C. $(x^3 + 4)(2x + 5)$

Explanation

The other options are incorrect because they do not accurately represent the factored form of the expression. Option A is missing the term $+4$ in the first factor, and option B is missing the term $+5$ in the second factor.

Step-by-Step Solution

Here are the step-by-step steps to solve this problem:

1. Identify the common factor among the terms in the given expression.
2. Factor out the common factor from the first two terms.
3. Factor out the common factor from the last two terms.
4. Combine the two expressions to get the final factored form.

Tips and Tricks

Here are some tips and tricks to help you solve this problem:

• Make sure to identify the common factor correctly.
• Factor out the common factor from each pair of terms separately.
• Combine the two expressions to get the final factored form.
• Check your answer by multiplying the two factors together to get the original expression.

Real-World Applications

Factoring out the common factor from an algebraic expression has many real-world applications. For example, it can be used to simplify complex equations, solve systems of equations, and optimize functions. It is also used in many fields such as physics, engineering, and economics.

Conclusion

In conclusion, factoring out the common factor from an algebraic expression is a crucial step in simplifying and solving equations. By identifying the common factor and factoring it out, we can express the given expression as a product of the common factor and another expression. This can be used to simplify complex equations, solve systems of equations, and optimize functions.
Q&A: Factoring Out the Common Factor

Q: What is factoring out the common factor?

A: Factoring out the common factor is a process of expressing an algebraic expression as a product of the common factor and another expression. This involves identifying the common factor among the terms in the given expression and factoring it out from each term.

Q: How do I identify the common factor?

A: To identify the common factor, you need to look for the term that divides each term in the expression without leaving a remainder. You can also use the distributive property to factor out the common factor.

Q: What is the distributive property?

A: The distributive property is a property of algebra that states that a single term can be distributed to multiple terms. For example, if you have the expression $a(b + c)$, you can distribute the term $a$ to each term inside the parentheses to get $ab + ac$.

Q: How do I factor out the common factor using the distributive property?

A: To factor out the common factor using the distributive property, you need to identify the common factor and distribute it to each term in the expression. For example, if you have the expression $2x^4 + 5x^3 - 8x - 20$, you can factor out the common factor $x^3$ by distributing it to each term to get $x^3(2x + 5) - 8x - 20$.

Q: What is the final factored form of the expression?

A: The final factored form of the expression is $(x^3 + 4)(2x + 5)$.

Q: How do I check my answer?

A: To check your answer, you can multiply the two factors together to get the original expression. For example, if you have the expression $(x^3 + 4)(2x + 5)$, you can multiply the two factors together to get $2x^4 + 5x^3 - 8x - 20$.

Q: What are some real-world applications of factoring out the common factor?

A: Factoring out the common factor has many real-world applications, including simplifying complex equations, solving systems of equations, and optimizing functions. It is also used in many fields such as physics, engineering, and economics.

Q: What are some common mistakes to avoid when factoring out the common factor?

A: Some common mistakes to avoid when factoring out the common factor include:

• Not identifying the common factor correctly
• Factoring out the wrong term
• Not distributing the common factor to each term
• Not checking the answer by multiplying the two factors together

Q: How can I practice factoring out the common factor?

A: You can practice factoring out the common factor by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Q: What are some tips and tricks for factoring out the common factor?

A: Some tips and tricks for factoring out the common factor include:

• Make sure to identify the common factor correctly
• Factor out the common factor from each pair of terms separately
• Combine the two expressions to get the final factored form
• Check your answer by multiplying the two factors together

Conclusion

In conclusion, factoring out the common factor is a crucial step in simplifying and solving equations. By identifying the common factor and factoring it out, we can express the given expression as a product of the common factor and another expression. This can be used to simplify complex equations, solve systems of equations, and optimize functions.