Enter The Correct Answer In The Box.What Is The Factored Form Of This Expression?${ 5p^3 - 10p^2 + 3p - 6 }$Write All Factors In Standard Form.

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Introduction

Factoring an expression is a fundamental concept in algebra that involves expressing a given expression as a product of simpler expressions, called factors. In this article, we will focus on factoring the given expression 5p3−10p2+3p−65p^3 - 10p^2 + 3p - 6 and write all factors in standard form.

Understanding the Expression

Before we begin factoring, let's take a closer look at the given expression. The expression is a polynomial of degree 3, which means it has three terms. The first term is 5p35p^3, the second term is −10p2-10p^2, the third term is 3p3p, and the last term is −6-6. Our goal is to factor this expression into simpler factors.

Factoring by Grouping

One of the most common methods of factoring is factoring by grouping. This method involves grouping the terms of the expression into pairs and then factoring out the greatest common factor (GCF) from each pair.

Step 1: Group the Terms

Let's group the terms of the expression into pairs:

5p3−10p25p^3 - 10p^2 and 3p−63p - 6

Step 2: Factor Out the GCF

Now, let's factor out the GCF from each pair:

5p3−10p2=5p2(p−2)5p^3 - 10p^2 = 5p^2(p - 2)

3p−6=3(p−2)3p - 6 = 3(p - 2)

Step 3: Combine the Factors

Now that we have factored out the GCF from each pair, we can combine the factors:

5p3−10p2+3p−6=5p2(p−2)+3(p−2)5p^3 - 10p^2 + 3p - 6 = 5p^2(p - 2) + 3(p - 2)

Step 4: Factor Out the Common Factor

Now, let's factor out the common factor (p−2)(p - 2) from both terms:

5p2(p−2)+3(p−2)=(5p2+3)(p−2)5p^2(p - 2) + 3(p - 2) = (5p^2 + 3)(p - 2)

Factoring by Greatest Common Factor (GCF)

Another method of factoring is factoring by GCF. This method involves factoring out the greatest common factor from all the terms of the expression.

Step 1: Identify the GCF

The GCF of the expression 5p3−10p2+3p−65p^3 - 10p^2 + 3p - 6 is 1.

Step 2: Factor Out the GCF

Since the GCF is 1, we cannot factor out any common factor from the expression.

Factoring by Difference of Squares

The expression 5p3−10p2+3p−65p^3 - 10p^2 + 3p - 6 cannot be factored using the difference of squares method.

Factoring by Sum and Difference

The expression 5p3−10p2+3p−65p^3 - 10p^2 + 3p - 6 cannot be factored using the sum and difference method.

Conclusion

In this article, we have factored the expression 5p3−10p2+3p−65p^3 - 10p^2 + 3p - 6 using the factoring by grouping method. We have also discussed other methods of factoring, including factoring by GCF, difference of squares, and sum and difference. However, the expression cannot be factored using these methods.

Final Answer

The factored form of the expression 5p3−10p2+3p−65p^3 - 10p^2 + 3p - 6 is:

(5p2+3)(p−2)(5p^2 + 3)(p - 2)

This is the final answer to the problem.

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Introduction

In our previous article, we discussed the factoring of the expression 5p3−10p2+3p−65p^3 - 10p^2 + 3p - 6 using the factoring by grouping method. 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&A

Q: What is factoring?

A: Factoring is the process of expressing a given expression as a product of simpler expressions, called factors.

Q: Why is factoring important?

A: Factoring is important because it allows us to simplify complex expressions and make them easier to work with. It also helps us to identify the underlying structure of an expression and make it easier to solve equations and inequalities.

Q: What are the different methods of factoring?

A: There are several methods of factoring, including:

  • Factoring by grouping
  • Factoring by GCF
  • Factoring by difference of squares
  • Factoring by sum and difference

Q: How do I choose the right method of factoring?

A: The choice of method depends on the type of expression you are working with. For example, if you have an expression that can be written as a product of two binomials, you may want to use the factoring by grouping method. If you have an expression that can be written as a difference of squares, you may want to use the factoring by difference of squares method.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides all the terms of an expression.

Q: How do I find the GCF of an expression?

A: To find the GCF of an expression, you can list all the factors of each term and then identify the largest factor that appears in all the lists.

Q: What is the difference of squares?

A: The difference of squares is a special type of expression that can be written as the difference of two squares.

Q: How do I factor a difference of squares?

A: To factor a difference of squares, you can use the formula:

a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

Q: What is the sum and difference?

A: The sum and difference is a special type of expression that can be written as the sum or difference of two terms.

Q: How do I factor a sum and difference?

A: To factor a sum and difference, you can use the formula:

a+b=(a+b)a + b = (a + b)

a−b=(a−b)a - b = (a - b)

Examples

Example 1: Factoring by Grouping

Factor the expression 2x2+5x+32x^2 + 5x + 3 using the factoring by grouping method.

Solution

2x2+5x+3=(2x2+3x)+(2x+3)2x^2 + 5x + 3 = (2x^2 + 3x) + (2x + 3)

=x(2x+3)+1(2x+3)= x(2x + 3) + 1(2x + 3)

=(2x+3)(x+1)= (2x + 3)(x + 1)

Example 2: Factoring by GCF

Factor the expression 6x2+12x+186x^2 + 12x + 18 using the factoring by GCF method.

Solution

6x2+12x+18=6(x2+2x+3)6x^2 + 12x + 18 = 6(x^2 + 2x + 3)

=6(x+1)(x+3)= 6(x + 1)(x + 3)

Example 3: Factoring by Difference of Squares

Factor the expression x2−4x^2 - 4 using the factoring by difference of squares method.

Solution

x2−4=(x+2)(x−2)x^2 - 4 = (x + 2)(x - 2)

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 provided examples of factoring by grouping, GCF, difference of squares, and sum and difference. By following these examples and practicing with different types of expressions, you will become proficient in factoring and be able to simplify complex expressions with ease.

Final Answer

The factored form of the expression 5p3−10p2+3p−65p^3 - 10p^2 + 3p - 6 is:

(5p2+3)(p−2)(5p^2 + 3)(p - 2)

This is the final answer to the problem.