Which Expression In Factored Form Is Equivalent To This Expression? 4\left(x^2-2x\right)-2\left(x^2-3\right ]A. 2 ( X − 1 ) ( X − 3 2(x-1)(x-3 2 ( X − 1 ) ( X − 3 ] B. ( 2 X + 3 ) ( X + 1 (2x+3)(x+1 ( 2 X + 3 ) ( X + 1 ] C. ( 2 X − 3 ) ( X + 1 (2x-3)(x+1 ( 2 X − 3 ) ( X + 1 ] D. 2 ( X + 1 ) ( X + 3 2(x+1)(x+3 2 ( X + 1 ) ( X + 3 ]

by ADMIN 336 views

Introduction

Factoring algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. In this article, we will explore the process of factoring algebraic expressions and apply it to a given problem. We will also discuss the importance of factoring in mathematics and provide tips for simplifying complex expressions.

What is Factoring?

Factoring is the process of expressing an algebraic expression as a product of simpler expressions. It involves breaking down an expression into its constituent parts and rewriting it in a more simplified form. Factoring is an essential skill in mathematics, as it allows us to simplify complex expressions, solve equations, and analyze functions.

Types of Factoring

There are several types of factoring, including:

  • Greatest Common Factor (GCF) Factoring: This involves factoring out the greatest common factor of two or more terms.
  • Difference of Squares Factoring: This involves factoring expressions of the form a2−b2a^2 - b^2.
  • Sum and Difference Factoring: This involves factoring expressions of the form a2+2ab+b2a^2 + 2ab + b^2 and a2−2ab+b2a^2 - 2ab + b^2.
  • Quadratic Factoring: This involves factoring quadratic expressions of the form ax2+bx+cax^2 + bx + c.

The Problem

The problem we will be working on is:

4(x2−2x)−2(x2−3)4\left(x^2-2x\right)-2\left(x^2-3\right)

We need to factor this expression and determine which of the given options is equivalent.

Step 1: Distribute the Coefficients

To factor the expression, we need to start by distributing the coefficients. This involves multiplying each term inside the parentheses by the coefficient outside the parentheses.

4(x2−2x)=4x2−8x4\left(x^2-2x\right) = 4x^2 - 8x

−2(x2−3)=−2x2+6-2\left(x^2-3\right) = -2x^2 + 6

Step 2: Combine Like Terms

Next, we need to combine like terms. This involves adding or subtracting terms that have the same variable and exponent.

4x2−8x−2x2+64x^2 - 8x - 2x^2 + 6

Combine like terms:

2x2−8x+62x^2 - 8x + 6

Step 3: Factor the Expression

Now that we have simplified the expression, we can factor it. We need to find two binomials whose product is equal to the expression.

Let's try to factor the expression by grouping:

2x2−8x+62x^2 - 8x + 6

We can factor out a 2 from the first two terms:

2(x2−4x)+62(x^2 - 4x) + 6

Now, we can factor out an x from the first term:

2x(x−4)+62x(x - 4) + 6

We can't factor the expression any further, so we need to try a different approach.

Step 4: Use the Distributive Property

Let's use the distributive property to rewrite the expression:

2x2−8x+62x^2 - 8x + 6

We can rewrite the expression as:

2x2−8x+6=2x(x−4)+62x^2 - 8x + 6 = 2x(x - 4) + 6

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

2x(x−4)+6=2x(x−4)+2(3)2x(x - 4) + 6 = 2x(x - 4) + 2(3)

We can factor out a 2 from the first two terms:

2(x(x−4)+3)2(x(x - 4) + 3)

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

2(x(x−4)+3)=2(x−1)(x−3)2(x(x - 4) + 3) = 2(x - 1)(x - 3)

Conclusion

In this article, we have explored the process of factoring algebraic expressions and applied it to a given problem. We have also discussed the importance of factoring in mathematics and provided tips for simplifying complex expressions. The correct answer to the problem is:

A. 2(x−1)(x−3)2(x-1)(x-3)

This is the only option that is equivalent to the given expression. The other options are not correct, as they do not factor the expression correctly.

Final Answer

The final answer is:

Introduction

Factoring algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. In our previous article, we explored the process of factoring algebraic expressions and applied it to a given problem. In this article, we will provide a Q&A guide to help you understand the concept of factoring and how to apply it to different types of expressions.

Q: What is factoring?

A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions. It involves breaking down an expression into its constituent parts and rewriting it in a more simplified form.

Q: Why is factoring important?

A: Factoring is an essential skill in mathematics, as it allows us to simplify complex expressions, solve equations, and analyze functions. It is also a crucial step in many mathematical operations, such as solving quadratic equations and finding the roots of a polynomial.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

  • Greatest Common Factor (GCF) Factoring: This involves factoring out the greatest common factor of two or more terms.
  • Difference of Squares Factoring: This involves factoring expressions of the form a2−b2a^2 - b^2.
  • Sum and Difference Factoring: This involves factoring expressions of the form a2+2ab+b2a^2 + 2ab + b^2 and a2−2ab+b2a^2 - 2ab + b^2.
  • Quadratic Factoring: This involves factoring quadratic expressions of the form ax2+bx+cax^2 + bx + c.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two binomials whose product is equal to the expression. You can use the following steps:

  1. Find the greatest common factor (GCF): If there is a GCF, factor it out of the expression.
  2. Look for two binomials: Try to find two binomials whose product is equal to the expression.
  3. Check your answer: Make sure that the product of the two binomials is equal to the original expression.

Q: What is the difference between factoring and simplifying?

A: Factoring and simplifying are two different mathematical operations. Factoring involves expressing an expression as a product of simpler expressions, while simplifying involves combining like terms and rewriting the expression in a more simplified form.

Q: Can you provide examples of factoring?

A: Yes, here are some examples of factoring:

  • Factoring a quadratic expression: x2+5x+6=(x+3)(x+2)x^2 + 5x + 6 = (x + 3)(x + 2)
  • Factoring a difference of squares: x2−4=(x−2)(x+2)x^2 - 4 = (x - 2)(x + 2)
  • Factoring a sum and difference: x2+2x+1=(x+1)2x^2 + 2x + 1 = (x + 1)^2
  • Factoring a greatest common factor: 6x2+12x=6x(x+2)6x^2 + 12x = 6x(x + 2)

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

A: Here are some common mistakes to avoid when factoring:

  • Not factoring out the greatest common factor: Make sure to factor out the greatest common factor of two or more terms.
  • Not checking your answer: Make sure that the product of the two binomials is equal to the original expression.
  • Not using the distributive property: Make sure to use the distributive property to rewrite the expression in a more simplified form.

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of factoring and how to apply it to different types of expressions. We have also discussed the importance of factoring in mathematics and provided tips for simplifying complex expressions. By following these tips and avoiding common mistakes, you can become proficient in factoring and simplify complex expressions with ease.

Final Answer

The final answer is:

Factoring is an essential skill in mathematics that involves expressing an expression as a product of simpler expressions. By understanding the different types of factoring and how to apply them, you can simplify complex expressions and solve equations with ease.