Factorize $7y + 21$.

Feb 26, 2025 by ADMIN 23 views

=====================================================

Introduction

In mathematics, factorization is a process of expressing an algebraic expression as a product of simpler expressions. It is an essential concept in algebra and is used to simplify complex expressions and solve equations. In this article, we will focus on factorizing the expression $7y + 21$.

Understanding the Expression

The given expression is $7y + 21$. This is a linear expression, which means it is of the form $ax + b$, where $a$ and $b$ are constants and $x$ is a variable. In this case, $a = 7$ and $b = 21$.

Factoring Out the Greatest Common Factor (GCF)

One of the most common methods of factorization is to factor out the greatest common factor (GCF) of the terms. The GCF is the largest expression that divides both terms without leaving a remainder. In this case, the GCF of $7y$ and $21$ is $7$.

To factor out the GCF, we need to divide both terms by the GCF. This can be done by dividing $7y$ by $7$, which gives us $y$, and dividing $21$ by $7$, which gives us $3$.

import sympy as sp

# Define the variables
y = sp.symbols('y')

# Define the expression
expr = 7*y + 21

# Factor out the GCF
gcf = 7
factored_expr = gcf * (y + 3)

print(factored_expr)

Factored Form

The factored form of the expression $7y + 21$ is $7(y + 3)$. This is a product of two simpler expressions: $7$ and $(y + 3)$.

Checking the Factored Form

To check if the factored form is correct, we can multiply the two expressions together and see if we get the original expression.

import sympy as sp

# Define the variables
y = sp.symbols('y')

# Define the factored expression
factored_expr = 7 * (y + 3)

# Multiply the expressions together
multiplied_expr = factored_expr.expand()

print(multiplied_expr)

Conclusion

Applications of Factorization

Factorization has many applications in mathematics and other fields. Some of the applications of factorization include:

Simplifying complex expressions: Factorization can be used to simplify complex expressions and make them easier to work with.
Solving equations: Factorization can be used to solve equations by setting each factor equal to zero and solving for the variable.
Graphing functions: Factorization can be used to graph functions by finding the x-intercepts of the function.
Optimization: Factorization can be used to optimize functions by finding the maximum or minimum value of the function.

Common Mistakes to Avoid

When factorizing expressions, there are several common mistakes to avoid. Some of these mistakes include:

Not factoring out the GCF: Failing to factor out the greatest common factor (GCF) can lead to incorrect factorization.
Factoring out the wrong term: Factoring out the wrong term can lead to incorrect factorization.
Not checking the factored form: Failing to check the factored form can lead to incorrect factorization.

Tips and Tricks

Here are some tips and tricks to help you factorize expressions:

Use the GCF method: The greatest common factor (GCF) method is one of the most common methods of factorization.
Use the difference of squares method: The difference of squares method is another common method of factorization.
Use the sum of cubes method: The sum of cubes method is a less common method of factorization, but it can be useful in certain situations.
Check your work: Always check your work to make sure that the factored form is correct.

Practice Problems

Here are some practice problems to help you practice factorizing expressions:

Factorize the expression $12x + 36$.
Factorize the expression $15y - 45$.
Factorize the expression $24z + 48$.

Conclusion

In this article, we factorized the expression $7y + 21$ using the method of factoring out the greatest common factor (GCF). We found that the factored form of the expression is $7(y + 3)$. We also checked the factored form by multiplying the two expressions together and verified that we get the original expression. We also discussed the applications of factorization, common mistakes to avoid, and tips and tricks to help you factorize expressions. Finally, we provided some practice problems to help you practice factorizing expressions.

=====================

Frequently Asked Questions

Q: What is factorization?

A: Factorization is the process of expressing an algebraic expression as a product of simpler expressions.

Q: Why is factorization important?

A: Factorization is important because it can be used to simplify complex expressions and make them easier to work with. It can also be used to solve equations and graph functions.

Q: What are the different methods of factorization?

A: There are several methods of factorization, including:

Factoring out the greatest common factor (GCF): This method involves factoring out the largest expression that divides both terms without leaving a remainder.
Difference of squares method: This method involves factoring expressions of the form $a^2 - b^2$ .
Sum of cubes method: This method involves factoring expressions of the form $a^3 + b^3$ .
Grouping method: This method involves grouping terms together and factoring out common factors.

Q: How do I factorize an expression?

A: To factorize an expression, you need to identify the greatest common factor (GCF) of the terms and factor it out. You can also use other methods such as the difference of squares method, sum of cubes method, or grouping method.

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

A: Some common mistakes to avoid when factorizing expressions include:

Not factoring out the GCF: Failing to factor out the greatest common factor (GCF) can lead to incorrect factorization.
Factoring out the wrong term: Factoring out the wrong term can lead to incorrect factorization.
Not checking the factored form: Failing to check the factored form can lead to incorrect factorization.

Q: How do I check if the factored form is correct?

A: To check if the factored form is correct, you need to multiply the two expressions together and see if you get the original expression.

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

A: Some real-world applications of factorization include:

Simplifying complex expressions: Factorization can be used to simplify complex expressions and make them easier to work with.
Solving equations: Factorization can be used to solve equations by setting each factor equal to zero and solving for the variable.
Graphing functions: Factorization can be used to graph functions by finding the x-intercepts of the function.
Optimization: Factorization can be used to optimize functions by finding the maximum or minimum value of the function.

Q: How do I practice factorizing expressions?

A: You can practice factorizing expressions by working through practice problems and exercises. You can also use online resources and tools to help you practice factorizing expressions.

Q: What are some resources for learning factorization?

A: Some resources for learning factorization include:

Textbooks: There are many textbooks available that cover factorization and other algebraic concepts.
Online resources: There are many online resources available that provide tutorials, examples, and practice problems for factorization.
Video tutorials: There are many video tutorials available that provide step-by-step instructions for factorization.
Practice problems: You can find practice problems for factorization in textbooks, online resources, and other places.

Conclusion

In this article, we answered some frequently asked questions about factorization. We discussed the importance of factorization, the different methods of factorization, and how to factorize expressions. We also discussed common mistakes to avoid, how to check if the factored form is correct, and some real-world applications of factorization. Finally, we provided some resources for learning factorization and practicing factorizing expressions.