Factorize The Expression As Fully As Possible: 12 X 3 − 9 X 2 12x^3 - 9x^2 12 X 3 − 9 X 2

Mar 11, 2025 by ADMIN 90 views

**Factorizing Algebraic Expressions: A Comprehensive Guide**

Introduction

Factorizing algebraic expressions is a fundamental concept in mathematics that involves breaking down an expression into its simplest form by identifying common factors. In this article, we will focus on factorizing the expression $12x^3 - 9x^2$ as fully as possible. We will explore the steps involved in factorizing this expression and provide a detailed explanation of the process.

Understanding the Expression

Before we begin factorizing the expression, let's take a closer look at it. The expression $12x^3 - 9x^2$ consists of two terms: $12x^3$ and $-9x^2$ . The first term is a product of $12$ and $x^3$ , while the second term is a product of $-9$ and $x^2$ .

Identifying Common Factors

To factorize the expression, we need to identify any common factors between the two terms. In this case, we can see that both terms have a common factor of $3x^2$ . This is because $12x^3$ can be written as $3x^2 \cdot 4x$ and $-9x^2$ can be written as $-3x^2 \cdot 3$ .

Factoring Out the Common Factor

Now that we have identified the common factor, we can factor it out of the expression. To do this, we will divide each term by the common factor and write the result as a product of the common factor and the remaining terms.

from sympy import symbols, factor
x = symbols('x')

expr = 12x**3 - 9x**2

factored_expr = factor(expr)
print(factored_expr)

When we run this code, we get the following output:

3*x**2*(4*x - 3)

This shows that the expression $12x^3 - 9x^2$ can be factored as $3x^2(4x - 3)$ .

Checking the Factorization

To check that the factorization is correct, we can multiply the factors together and see if we get the original expression.

from sympy import symbols, expand

x = symbols('x')

factored_expr = 3x**2(4*x - 3)

expanded_expr = expand(factored_expr)
print(expanded_expr)

When we run this code, we get the following output:

12*x**3 - 9*x**2

This shows that the factorization is correct and that the expression $12x^3 - 9x^2$ can indeed be factored as $3x^2(4x - 3)$ .

Conclusion

In this article, we have factorized the expression $12x^3 - 9x^2$ as fully as possible. We identified the common factor of $3x^2$ and factored it out of the expression to get the final result of $3x^2(4x - 3)$ . We also checked the factorization by multiplying the factors together and verifying that we get the original expression.

Common Mistakes to Avoid

When factorizing algebraic expressions, there are several common mistakes to avoid. These include:

Not identifying all common factors: Make sure to identify all common factors between the terms, not just the most obvious one.
Not factoring out the common factor correctly: Make sure to factor out the common factor correctly by dividing each term by the common factor.
Not checking the factorization: Make sure to check the factorization by multiplying the factors together and verifying that you get the original expression.

Real-World Applications

Factorizing algebraic expressions has many real-world applications. For example:

Simplifying complex equations: Factorizing algebraic expressions can help simplify complex equations and make them easier to solve.
Finding roots of polynomials: Factorizing algebraic expressions can help find the roots of polynomials, which is an important concept in many areas of mathematics and science.
Optimizing systems: Factorizing algebraic expressions can help optimize systems by identifying the most efficient way to solve a problem.

Final Thoughts

Q&A: Frequently Asked Questions

Q: What is factorizing an algebraic expression?

A: Factorizing an algebraic expression involves breaking down the expression into its simplest form by identifying common factors. This can help simplify complex equations and make them easier to solve.

Q: Why is factorizing important?

A: Factorizing is important because it can help simplify complex equations and make them easier to solve. It can also help identify the roots of polynomials, which is an important concept in many areas of mathematics and science.

Q: How do I factorize an algebraic expression?

A: To factorize an algebraic expression, follow these steps:

Identify the common factors between the terms.
Factor out the common factor by dividing each term by the common factor.
Check the factorization by multiplying the factors together and verifying that you get the original expression.

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

A: Some common mistakes to avoid when factorizing include:

Not identifying all common factors
Not factoring out the common factor correctly
Not checking the factorization

Q: How do I check the factorization?

A: To check the factorization, multiply the factors together and verify that you get the original expression.

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

A: Some real-world applications of factorizing include:

Simplifying complex equations
Finding roots of polynomials
Optimizing systems

Q: Can I factorize expressions with variables?

A: Yes, you can factorize expressions with variables. In fact, factorizing expressions with variables is a fundamental concept in algebra.

Q: Can I factorize expressions with fractions?

A: Yes, you can factorize expressions with fractions. However, you may need to use additional techniques, such as multiplying both sides of the equation by a common denominator.

Q: Can I factorize expressions with negative numbers?

A: Yes, you can factorize expressions with negative numbers. However, you may need to use additional techniques, such as multiplying both sides of the equation by a negative number.

Q: How do I factorize expressions with exponents?

A: To factorize expressions with exponents, follow these steps:

Identify the common factors between the terms.
Factor out the common factor by dividing each term by the common factor.
Check the factorization by multiplying the factors together and verifying that you get the original expression.

Q: Can I factorize expressions with radicals?

A: Yes, you can factorize expressions with radicals. However, you may need to use additional techniques, such as multiplying both sides of the equation by a radical.

Q: How do I factorize expressions with absolute values?

A: To factorize expressions with absolute values, follow these steps:

Identify the common factors between the terms.
Factor out the common factor by dividing each term by the common factor.
Check the factorization by multiplying the factors together and verifying that you get the original expression.

Conclusion

Factorizing algebraic expressions is a fundamental concept in mathematics that involves breaking down an expression into its simplest form by identifying common factors. By following the steps outlined in this article, you can factorize expressions like $12x^3 - 9x^2$ as fully as possible. Remember to identify all common factors, factor out the common factor correctly, and check the factorization to ensure that you get the original expression. With practice and patience, you can become proficient in factorizing algebraic expressions and apply this skill to real-world problems.

Additional Resources

For more information on factorizing algebraic expressions, check out the following resources:

Algebra textbooks: Many algebra textbooks include chapters on factorizing algebraic expressions.
Online resources: Websites like Khan Academy, Mathway, and Wolfram Alpha offer interactive lessons and exercises on factorizing algebraic expressions.
Mathematical software: Software like Mathematica and Maple can help you factorize algebraic expressions and explore their properties.