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Mar 13, 2025 by ADMIN 192 views

**What is $12x^3 - 9x^2 - 4x + 3$ in Factored Form?**

Introduction

Factoring polynomials is a crucial concept in algebra, and it plays a vital role in solving equations and inequalities. In this article, we will focus on factoring a given polynomial expression, $12x^3 - 9x^2 - 4x + 3$ . We will use various factoring techniques to simplify the expression and present it in its factored form.

Understanding the Polynomial Expression

Before we begin factoring, let's take a closer look at the given polynomial expression:

$12x^3 - 9x^2 - 4x + 3$

This expression consists of four terms, each with a different degree. The first term, $12x^3$ , has a degree of 3, while the second term, $-9x^2$ , has a degree of 2. The third term, $-4x$ , has a degree of 1, and the fourth term, $3$ , has a degree of 0.

Factoring by Grouping

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

Let's apply this technique to the given polynomial expression:

$12x^3 - 9x^2 - 4x + 3$

We can group the first two terms and the last two terms as follows:

$(12x^3 - 9x^2) - (4x - 3)$

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

$3x^2(4x - 3) - 1(4x - 3)$

We can see that both pairs have a common factor of $(4x - 3)$ . Let's factor this out:

$(3x^2 - 1)(4x - 3)$

Factoring the Quadratic Expression

The expression $3x^2 - 1$ is a quadratic expression that can be factored using the difference of squares formula:

$a^2 - b^2 = (a + b)(a - b)$

In this case, we have:

$3x^2 - 1 = (3x)^2 - 1^2$

Applying the difference of squares formula, we get:

$(3x)^2 - 1^2 = (3x + 1)(3x - 1)$

So, the factored form of the quadratic expression is:

$(3x + 1)(3x - 1)$

Combining the Factored Forms

Now that we have factored the quadratic expression, we can combine the factored forms of the two expressions:

$(3x^2 - 1)(4x - 3) = (3x + 1)(3x - 1)(4x - 3)$

This is the final factored form of the given polynomial expression.

Conclusion

In this article, we have factored the polynomial expression $12x^3 - 9x^2 - 4x + 3$ using various factoring techniques. We have applied the factoring by grouping method and factored the quadratic expression using the difference of squares formula. The final factored form of the expression is $(3x + 1)(3x - 1)(4x - 3)$ . This expression can be used to simplify complex equations and inequalities involving the given polynomial.

Common Mistakes to Avoid

When factoring polynomial expressions, it's essential to avoid common mistakes that can lead to incorrect results. Here are some common mistakes to avoid:

Not factoring out the GCF: Failing to factor out the greatest common factor (GCF) from each pair of terms can lead to incorrect results.
Not using the correct factoring technique: Using the wrong factoring technique can lead to incorrect results.
Not checking for common factors: Failing to check for common factors between terms can lead to incorrect results.

Tips and Tricks

Here are some tips and tricks to help you factor polynomial expressions:

Use the factoring by grouping method: This method involves grouping the terms of the polynomial expression into pairs and then factoring out the GCF from each pair.
Use the difference of squares formula: This formula can be used to factor quadratic expressions that can be written in the form $a^2 - b^2$ .
Check for common factors: Failing to check for common factors between terms can lead to incorrect results.

Real-World Applications

Factoring polynomial expressions has numerous real-world applications in various fields, including:

Engineering: Factoring polynomial expressions is used to solve complex equations and inequalities involving polynomial functions.
Physics: Factoring polynomial expressions is used to solve complex equations and inequalities involving polynomial functions in physics.
Computer Science: Factoring polynomial expressions is used to solve complex equations and inequalities involving polynomial functions in computer science.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about factoring polynomial expressions.

Q: What is factoring in algebra?

A: Factoring in algebra involves expressing a polynomial expression as a product of simpler expressions, called factors. Factoring is a crucial concept in algebra that helps us simplify complex equations and inequalities involving polynomial functions.

Q: What are the different factoring techniques?

A: There are several factoring techniques, including:

Factoring by grouping: This method involves grouping the terms of the polynomial expression into pairs and then factoring out the greatest common factor (GCF) from each pair.
Factoring out the GCF: This method involves factoring out the greatest common factor (GCF) from each term of the polynomial expression.
Factoring quadratic expressions: This method involves factoring quadratic expressions that can be written in the form $a^2 - b^2$ .
Factoring by difference of squares: This method involves factoring quadratic expressions that can be written in the form $a^2 - b^2$ .

Q: How do I factor a polynomial expression?

A: To factor a polynomial expression, follow these steps:

Check for common factors: Check if there are any common factors between the terms of the polynomial expression.
Use the factoring by grouping method: Group the terms of the polynomial expression into pairs and then factor out the greatest common factor (GCF) from each pair.
Use the factoring out the GCF method: Factor out the greatest common factor (GCF) from each term of the polynomial expression.
Use the factoring quadratic expressions method: Factor quadratic expressions that can be written in the form $a^2 - b^2$ .
Use the factoring by difference of squares method: Factor quadratic expressions that can be written in the form $a^2 - b^2$ .

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

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

Not factoring out the GCF: Failing to factor out the greatest common factor (GCF) from each pair of terms can lead to incorrect results.
Not using the correct factoring technique: Using the wrong factoring technique can lead to incorrect results.
Not checking for common factors: Failing to check for common factors between terms can lead to incorrect results.

Q: How do I check for common factors?

A: To check for common factors, follow these steps:

List the factors of each term: List the factors of each term of the polynomial expression.
Find the common factors: Find the common factors between the terms of the polynomial expression.
Factor out the common factors: Factor out the common factors from each term of the polynomial expression.

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

A: Factoring polynomial expressions has numerous real-world applications in various fields, including:

Engineering: Factoring polynomial expressions is used to solve complex equations and inequalities involving polynomial functions.
Physics: Factoring polynomial expressions is used to solve complex equations and inequalities involving polynomial functions in physics.
Computer Science: Factoring polynomial expressions is used to solve complex equations and inequalities involving polynomial functions in computer science.

Conclusion

In conclusion, factoring polynomial expressions is a crucial concept in algebra that has numerous real-world applications. By understanding the various factoring techniques and avoiding common mistakes, you can simplify complex equations and inequalities involving polynomial functions.