Factor: $3x^2 + 4x + 21x + 28$A. $(7x + 3)(x + 4$\] B. $(3x + 7)(x + 4$\] C. $(3x + 4)(x + 7$\] D. $x(3x + 4 + 21) + 28$

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two or more polynomials. In this article, we will explore the process of factoring quadratic expressions and provide a step-by-step guide on how to factor a given quadratic expression.

What is Factoring?

Factoring is the process of expressing a quadratic expression as a product of two or more polynomials. It involves finding the factors of the quadratic expression and expressing it in the form of a product of these factors. Factoring is an essential concept in algebra and is used to solve quadratic equations, simplify expressions, and find the roots of a quadratic equation.

Types of Factoring

There are several types of factoring, including:

  • Factoring by Grouping: This involves grouping the terms of the quadratic expression into two groups and then factoring out the common factors from each group.
  • Factoring by Difference of Squares: This involves factoring a quadratic expression that can be expressed as the difference of two squares.
  • Factoring by Perfect Square Trinomials: This involves factoring a quadratic expression that can be expressed as a perfect square trinomial.

Factoring the Given Quadratic Expression

The given quadratic expression is 3x2+4x+21x+283x^2 + 4x + 21x + 28. To factor this expression, we need to first group the terms and then factor out the common factors.

Step 1: Group the Terms

The given quadratic expression can be grouped as follows:

3x2+4x+21x+28=(3x2+21x)+(4x+28)3x^2 + 4x + 21x + 28 = (3x^2 + 21x) + (4x + 28)

Step 2: Factor Out the Common Factors

Now, we need to factor out the common factors from each group.

(3x2+21x)=3x(x+7)(3x^2 + 21x) = 3x(x + 7)

(4x+28)=4(x+7)(4x + 28) = 4(x + 7)

Step 3: Combine the Factored Groups

Now, we can combine the factored groups to get the final factored form of the quadratic expression.

3x2+4x+21x+28=3x(x+7)+4(x+7)3x^2 + 4x + 21x + 28 = 3x(x + 7) + 4(x + 7)

Step 4: Factor Out the Common Binomial Factor

Now, we can factor out the common binomial factor from the two groups.

3x(x+7)+4(x+7)=(3x+4)(x+7)3x(x + 7) + 4(x + 7) = (3x + 4)(x + 7)

Conclusion

In this article, we have explored the process of factoring quadratic expressions and provided a step-by-step guide on how to factor a given quadratic expression. We have also discussed the different types of factoring and how to factor a quadratic expression using the difference of squares and perfect square trinomials. The final factored form of the given quadratic expression is (3x+4)(x+7)(3x + 4)(x + 7).

Final Answer

The final answer is: (3x+4)(x+7)\boxed{(3x + 4)(x + 7)}

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that involves expressing a quadratic expression as a product of two or more polynomials. In this article, we will provide a Q&A guide on factoring quadratic expressions, covering common questions and topics related to factoring.

Q&A: Factoring Quadratic Expressions

Q: What is factoring in algebra?

A: Factoring is the process of expressing a quadratic expression as a product of two or more polynomials. It involves finding the factors of the quadratic expression and expressing it in the form of a product of these factors.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Factoring by Grouping: This involves grouping the terms of the quadratic expression into two groups and then factoring out the common factors from each group.
  • Factoring by Difference of Squares: This involves factoring a quadratic expression that can be expressed as the difference of two squares.
  • Factoring by Perfect Square Trinomials: This involves factoring a quadratic expression that can be expressed as a perfect square trinomial.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to follow these steps:

  1. Group the terms: Group the terms of the quadratic expression into two groups.
  2. Factor out the common factors: Factor out the common factors from each group.
  3. Combine the factored groups: Combine the factored groups to get the final factored form of the quadratic expression.
  4. Factor out the common binomial factor: Factor out the common binomial factor from the two groups.

Q: What is the difference between factoring and simplifying?

A: Factoring and simplifying are two different concepts in algebra. Factoring involves expressing a quadratic expression as a product of two or more polynomials, while simplifying involves reducing a quadratic expression to its simplest form.

Q: How do I determine if a quadratic expression can be factored?

A: To determine if a quadratic expression can be factored, you need to check if it can be expressed as a product of two or more polynomials. You can use the following methods to check:

  • Check if the quadratic expression can be expressed as a difference of squares: If the quadratic expression can be expressed as a difference of squares, it can be factored using the difference of squares formula.
  • Check if the quadratic expression can be expressed as a perfect square trinomial: If the quadratic expression can be expressed as a perfect square trinomial, it can be factored using the perfect square trinomial formula.

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

A: Some common mistakes to avoid when factoring quadratic expressions include:

  • Not grouping the terms correctly: Make sure to group the terms correctly before factoring.
  • Not factoring out the common factors correctly: Make sure to factor out the common factors correctly from each group.
  • Not combining the factored groups correctly: Make sure to combine the factored groups correctly to get the final factored form of the quadratic expression.

Conclusion

In this article, we have provided a Q&A guide on factoring quadratic expressions, covering common questions and topics related to factoring. We have also discussed the different types of factoring and how to determine if a quadratic expression can be factored. By following the steps outlined in this article, you can master the art of factoring quadratic expressions and become proficient in algebra.

Final Answer

The final answer is: (3x+4)(x+7)\boxed{(3x + 4)(x + 7)}