Factor Completely:$\[ 7w^9 - 49w^2 + 6w - 42 \\]

Feb 28, 2025 by ADMIN 49 views

**Factor Completely: A Comprehensive Guide to Algebraic Expressions**

Introduction

Factoring algebraic expressions is a fundamental concept in mathematics, and it plays a crucial role in solving equations and inequalities. In this article, we will focus on factoring the given expression: $7w^9 - 49w^2 + 6w - 42$ . We will break down the expression into its prime factors and simplify it to its simplest form.

Understanding the Expression

Before we start factoring, let's take a closer look at the given expression: $7w^9 - 49w^2 + 6w - 42$ . This expression consists of four terms, and we can see that it has two common factors: $7w^9$ and $-49w^2$ . However, we can also notice that the last two terms, $6w$ and $-42$ , have a common factor of $6$ .

Factoring by Grouping

One of the most effective ways to factor an expression is by grouping. This method involves grouping the terms of the expression into pairs and then factoring out the common factors from each pair. Let's apply this method to the given expression.

7w^9 - 49w^2 + 6w - 42
= (7w^9 - 49w^2) + (6w - 42)
= 7w^9 - 49w^2 + 6w - 42

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

= 7w^9 - 49w^2 + 6w - 42
= 7w^9 - 7w^2(7w) + 6w - 6(7)
= 7w^9 - 7w^2(7w) + 6w - 6(7)

We can see that the first pair of terms has a common factor of $7w^9$ , and the second pair of terms has a common factor of $6$ . Let's factor out these common factors:

= 7w^9 - 7w^2(7w) + 6w - 6(7)
= 7w^9(1 - 7w) + 6(1 - 7w)

Now, we can see that both pairs of terms have a common factor of $(1 - 7w)$ . Let's factor this out:

= 7w^9(1 - 7w) + 6(1 - 7w)
= (1 - 7w)(7w^9 + 6)

Factoring by Difference of Squares

We can see that the expression $(7w^9 + 6)$ can be factored using the difference of squares formula. However, we need to rewrite the expression in the form of a difference of squares. Let's rewrite the expression as follows:

(7w^9 + 6) = (7w^4)^2 + 6

Now, we can see that the expression is in the form of a difference of squares. Let's apply the difference of squares formula:

= (7w^4)^2 + 6
= (7w^4 + √6)(7w^4 - √6)

However, we can see that the expression $(7w^4 - √6)$ cannot be factored further using the difference of squares formula. Therefore, we can conclude that the expression $(7w^9 + 6)$ cannot be factored further.

Factoring by Grouping (Alternative Method)

Let's try an alternative method to factor the expression $(7w^9 + 6)$ . We can group the terms of the expression into pairs and then factor out the common factors from each pair.

(7w^9 + 6) = (7w^9 + 0) + (0 + 6)
= 7w^9 + 6

We can see that the first pair of terms has a common factor of $7w^9$ , and the second pair of terms has a common factor of $6$ . Let's factor out these common factors:

= 7w^9 + 6
= 7w^9 + 6

We can see that the expression $(7w^9 + 6)$ cannot be factored further using this method.

Conclusion

In this article, we have factored the given expression: $7w^9 - 49w^2 + 6w - 42$ . We have used the method of factoring by grouping to factor the expression into its prime factors. We have also tried an alternative method to factor the expression, but it did not yield any further simplification.

Final Answer

The final answer is:

\boxed{(1 - 7w)(7w^9 + 6)}

Note

Q: What is factoring in algebra?

A: Factoring in algebra is the process of expressing an algebraic expression as a product of simpler expressions, called factors. This is done by identifying the common factors of the terms in the expression and factoring them out.

Q: Why is factoring important in algebra?

A: Factoring is an essential concept in algebra because it allows us to simplify complex expressions and solve equations and inequalities. By factoring an expression, we can identify its roots and solve for the unknown variable.

Q: What are the different methods of factoring?

A: There are several methods of factoring, including:

Factoring by grouping
Factoring by difference of squares
Factoring by sum and difference of cubes
Factoring by greatest common factor (GCF)

Q: What is factoring by grouping?

A: Factoring by grouping is a method of factoring where we group the terms of the expression into pairs and then factor out the common factors from each pair.

Q: What is factoring by difference of squares?

A: Factoring by difference of squares is a method of factoring where we use the formula $a^2 - b^2 = (a + b)(a - b)$ to factor the expression.

Q: What is factoring by sum and difference of cubes?

A: Factoring by sum and difference of cubes is a method of factoring where we use the formulas $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ and $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ to factor the expression.

Q: What is factoring by greatest common factor (GCF)?

A: Factoring by greatest common factor (GCF) is a method of factoring where we identify the greatest common factor of the terms in the expression and factor it out.

Q: How do I know which method to use?

A: To determine which method to use, you need to analyze the expression and identify the common factors. If the expression can be grouped into pairs, you can use factoring by grouping. If the expression is in the form of a difference of squares, you can use factoring by difference of squares. If the expression is in the form of a sum or difference of cubes, you can use factoring by sum and difference of cubes. If the expression has a common factor, you can use factoring by GCF.

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

A: Some common mistakes to avoid when factoring include:

Not identifying the common factors
Not grouping the terms correctly
Not using the correct formula for factoring by difference of squares or sum and difference of cubes
Not factoring out the greatest common factor (GCF)

Q: How do I check my work when factoring?

A: To check your work when factoring, you need to multiply the factors together and see if you get the original expression. If you do, then your factoring is correct. If you don't, then you need to go back and recheck your work.

Q: What are some real-world applications of factoring?

A: Factoring has many real-world applications, including:

Solving equations and inequalities in physics and engineering
Modeling population growth and decline in biology
Analyzing data in statistics
Solving optimization problems in economics

Conclusion

Factoring is an essential concept in algebra that allows us to simplify complex expressions and solve equations and inequalities. By understanding the different methods of factoring and how to apply them, we can solve a wide range of problems in mathematics and other fields.