Factor The Polynomial: 2 X ( X − 4 ) + 7 ( X − 4 2x(x-4) + 7(x-4 2 X ( X − 4 ) + 7 ( X − 4 ]A. ( X − 4 ) ( 2 X − 7 (x-4)(2x-7 ( X − 4 ) ( 2 X − 7 ] B. ( X − 4 ) ( 2 X + 7 (x-4)(2x+7 ( X − 4 ) ( 2 X + 7 ] C. 14 X ( X − 4 14x(x-4 14 X ( X − 4 ] D. ( 2 X − 4 ) ( X + 7 (2x-4)(x+7 ( 2 X − 4 ) ( X + 7 ]

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 focus on factoring the given polynomial $2x(x-4) + 7(x-4)$ and explore the different methods and techniques used to factor polynomials.

What is Factoring?

Factoring a polynomial involves expressing it as a product of two or more polynomials. This can be done in various ways, depending on the type of polynomial and its structure. Factoring polynomials is an essential skill in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions.

The Given Polynomial

The given polynomial is $2x(x-4) + 7(x-4)$. To factor this polynomial, we need to identify the common factor and use the distributive property to simplify the expression.

Step 1: Identify the Common Factor

The common factor in the given polynomial is $(x-4)$. This factor is present in both terms of the polynomial.

Step 2: Use the Distributive Property

Using the distributive property, we can rewrite the polynomial as:

$2x(x-4) + 7(x-4) = (2x+7)(x-4)$

Step 3: Factor the Polynomial

Now that we have identified the common factor and used the distributive property, we can factor the polynomial as:

$(2x+7)(x-4)$

Comparing the Factored Form

The factored form of the polynomial is $(2x+7)(x-4)$. Let's compare this with the given options:

A. $(x-4)(2x-7)$ B. $(x-4)(2x+7)$ C. $14x(x-4)$ D. $(2x-4)(x+7)$

Conclusion

Based on our analysis, the correct factored form of the polynomial is:

$(2x+7)(x-4)$

This matches option B. $(x-4)(2x+7)$.

Why is Factoring Important?

Factoring polynomials is an essential skill in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions. By factoring polynomials, we can:

• Simplify complex expressions
• Solve equations
• Analyze functions
• Identify patterns and relationships between variables

Common Mistakes to Avoid

When factoring polynomials, it's essential to avoid common mistakes such as:

• Not identifying the common factor
• Not using the distributive property
• Not factoring the polynomial correctly

Tips and Tricks

Here are some tips and tricks to help you factor polynomials:

• Identify the common factor first
• Use the distributive property to simplify the expression
• Factor the polynomial correctly
• Check your work by plugging in values

Conclusion

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. By following the steps outlined in this article, you can factor polynomials with ease and accuracy. Remember to identify the common factor, use the distributive property, and factor the polynomial correctly. With practice and patience, you'll become a pro at factoring polynomials in no time!

Additional Resources

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

Practice Problems

• Factor the polynomial $3x(x+2) + 5(x+2)$
• Factor the polynomial $2x(x-3) + 4(x-3)$
• Factor the polynomial $x(x+1) + 2(x+1)$

Solutions

• $3x(x+2) + 5(x+2) = (3x+5)(x+2)$
• $2x(x-3) + 4(x-3) = (2x+4)(x-3)$
• $x(x+1) + 2(x+1) = (x+2)(x+1)$
  Factoring Polynomials: A Q&A Guide =====================================

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 answer some of the most frequently asked questions about factoring polynomials.

Q: What is factoring?

A: Factoring a polynomial involves expressing it as a product of two or more polynomials. This can be done in various ways, depending on the type of polynomial and its structure.

Q: Why is factoring important?

A: Factoring polynomials is an essential skill in algebra, as it allows us to simplify complex expressions, solve equations, and analyze functions. By factoring polynomials, we can:

• Simplify complex expressions
• Solve equations
• Analyze functions
• Identify patterns and relationships between variables

Q: How do I factor a polynomial?

A: To factor a polynomial, you need to identify the common factor and use the distributive property to simplify the expression. Here are the steps to follow:

1. Identify the common factor
2. Use the distributive property to simplify the expression
3. Factor the polynomial correctly

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

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

• Not identifying the common factor
• Not using the distributive property
• Not factoring the polynomial correctly

Q: How do I identify the common factor?

A: To identify the common factor, you need to look for the term that is present in both terms of the polynomial. This term is the common factor.

Q: What is the distributive property?

A: The distributive property is a mathematical property that allows us to multiply a single term by multiple terms. It states that:

a(b + c) = ab + ac

Q: How do I use the distributive property to simplify an expression?

A: To use the distributive property to simplify an expression, you need to multiply the single term by each of the multiple terms. Here are the steps to follow:

1. Multiply the single term by the first term
2. Multiply the single term by the second term
3. Add the results together

Q: What are some tips and tricks for factoring polynomials?

A: Here are some tips and tricks for factoring polynomials:

• Identify the common factor first
• Use the distributive property to simplify the expression
• Factor the polynomial correctly
• Check your work by plugging in values

Q: How do I check my work when factoring polynomials?

A: To check your work when factoring polynomials, you need to plug in values to see if the factored form is correct. Here are the steps to follow:

1. Plug in a value for the variable
2. Simplify the expression
3. Check if the result is correct

Q: What are some common types of polynomials that can be factored?

A: Some common types of polynomials that can be factored include:

• Quadratic polynomials
• Cubic polynomials
• Quartic polynomials

Q: How do I factor quadratic polynomials?

A: To factor quadratic polynomials, you need to use the factoring method. Here are the steps to follow:

1. Identify the common factor
2. Use the factoring method to simplify the expression
3. Factor the polynomial correctly

Q: How do I factor cubic polynomials?

A: To factor cubic polynomials, you need to use the factoring method. Here are the steps to follow:

1. Identify the common factor
2. Use the factoring method to simplify the expression
3. Factor the polynomial correctly

Q: How do I factor quartic polynomials?

A: To factor quartic polynomials, you need to use the factoring method. Here are the steps to follow:

1. Identify the common factor
2. Use the factoring method to simplify the expression
3. Factor the polynomial correctly

Conclusion

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. By following the steps outlined in this article, you can factor polynomials with ease and accuracy. Remember to identify the common factor, use the distributive property, and factor the polynomial correctly. With practice and patience, you'll become a pro at factoring polynomials in no time!

Additional Resources

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

Practice Problems

• Factor the polynomial $3x(x+2) + 5(x+2)$
• Factor the polynomial $2x(x-3) + 4(x-3)$
• Factor the polynomial $x(x+1) + 2(x+1)$

Solutions

• $3x(x+2) + 5(x+2) = (3x+5)(x+2)$
• $2x(x-3) + 4(x-3) = (2x+4)(x-3)$
• $x(x+1) + 2(x+1) = (x+2)(x+1)$