Factorise The Following Algebraic Expression: $ 8n(y-4) + 5(y-4) }$Answer { \square$ $

Mar 8, 2025 by ADMIN 88 views

**Factorising Algebraic Expressions: A Step-by-Step Guide**

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Introduction

Factorising algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. This technique is essential in solving equations, simplifying expressions, and understanding the properties of functions. In this article, we will focus on factorising the given algebraic expression: $8n(y-4) + 5(y-4)$ .

Understanding the Expression

The given expression is a linear combination of two terms, each of which contains the factor $(y-4)$ . To factorise this expression, we need to identify the common factor and then use the distributive property to simplify the expression.

The Distributive Property

The distributive property states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

$a(b+c) = ab + ac$

This property allows us to expand and simplify expressions by distributing the terms.

Factorising the Expression

To factorise the given expression, we can use the distributive property to expand the terms and then identify the common factor.

$8n(y-4) + 5(y-4)$

Using the distributive property, we can expand the terms as follows:

$8n(y-4) = 8ny - 32n$

$5(y-4) = 5y - 20$

Now, we can combine the two terms:

$8ny - 32n + 5y - 20$

Identifying the Common Factor

The common factor in the expression is $(y-4)$ . We can factor this out using the distributive property:

$(y-4)(8n + 5)$

Simplifying the Expression

The expression is now simplified, and we can see that the common factor $(y-4)$ has been factored out.

Conclusion

Factorising algebraic expressions is an essential technique in mathematics that involves expressing an expression as a product of simpler expressions. By identifying the common factor and using the distributive property, we can simplify expressions and understand the properties of functions. In this article, we have factorised the given algebraic expression: $8n(y-4) + 5(y-4)$ , and the simplified expression is $(y-4)(8n + 5)$ .

Real-World Applications

Factorising algebraic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, factorising expressions is used to describe the motion of objects and to calculate forces and energies. In engineering, factorising expressions is used to design and optimize systems, such as electrical circuits and mechanical systems. In economics, factorising expressions is used to model and analyze economic systems, such as supply and demand curves.

Tips and Tricks

Here are some tips and tricks to help you factorise algebraic expressions:

Identify the common factor: The first step in factorising an expression is to identify the common factor. This can be a variable, a constant, or a combination of both.
Use the distributive property: The distributive property is a powerful tool for factorising expressions. It allows us to expand and simplify expressions by distributing the terms.
Simplify the expression: Once you have identified the common factor, you can simplify the expression by combining the terms.
Check your work: Finally, it is essential to check your work to ensure that the expression has been factorised correctly.

Common Mistakes

Here are some common mistakes to avoid when factorising algebraic expressions:

Not identifying the common factor: Failing to identify the common factor is a common mistake when factorising expressions.
Not using the distributive property: Failing to use the distributive property can lead to incorrect factorisation.
Not simplifying the expression: Failing to simplify the expression can lead to unnecessary complexity.
Not checking your work: Failing to check your work can lead to errors and incorrect factorisation.

Conclusion

Factorising algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. By identifying the common factor and using the distributive property, we can simplify expressions and understand the properties of functions. In this article, we have factorised the given algebraic expression: $8n(y-4) + 5(y-4)$ , and the simplified expression is $(y-4)(8n + 5)$ . We have also discussed the real-world applications of factorising algebraic expressions, provided tips and tricks, and highlighted common mistakes to avoid.

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Introduction

Factorising algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. In our previous article, we discussed the basics of factorising algebraic expressions and provided a step-by-step guide on how to factorise the expression $8n(y-4) + 5(y-4)$ . In this article, we will answer some frequently asked questions about factorising algebraic expressions.

Q&A

Q: What is factorising an algebraic expression?

A: Factorising an algebraic expression involves expressing the expression as a product of simpler expressions. This is done by identifying the common factor and using the distributive property to simplify the expression.

Q: How do I identify the common factor in an algebraic expression?

A: To identify the common factor, look for the term that is common to all the terms in the expression. This can be a variable, a constant, or a combination of both.

Q: What is the distributive property?

A: The distributive property is a mathematical property that states that for any real numbers $a$ , $b$ , and $c$ , the following equation holds:

$a(b+c) = ab + ac$

This property allows us to expand and simplify expressions by distributing the terms.

Q: How do I use the distributive property to factorise an algebraic expression?

A: To use the distributive property, first identify the common factor and then use the distributive property to expand the terms. For example, if we have the expression $8n(y-4) + 5(y-4)$ , we can use the distributive property to expand the terms as follows:

$8n(y-4) = 8ny - 32n$

$5(y-4) = 5y - 20$

Now, we can combine the two terms:

$8ny - 32n + 5y - 20$

Q: What is the difference between factoring and simplifying an algebraic expression?

A: Factoring an algebraic expression involves expressing the expression as a product of simpler expressions, while simplifying an algebraic expression involves combining like terms to reduce the expression to its simplest form.

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, combine like terms by adding or subtracting the coefficients of the terms. For example, if we have the expression $8n(y-4) + 5(y-4)$ , we can simplify it by combining the like terms as follows:

$(y-4)(8n + 5)$

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

A: Some common mistakes to avoid when factorising algebraic expressions include:

Not identifying the common factor
Not using the distributive property
Not simplifying the expression
Not checking your work

Conclusion

Factorising algebraic expressions is a fundamental concept in mathematics that involves expressing an expression as a product of simpler expressions. By identifying the common factor and using the distributive property, we can simplify expressions and understand the properties of functions. In this article, we have answered some frequently asked questions about factorising algebraic expressions and provided tips and tricks to help you factorise expressions correctly.

Tips and Tricks

Here are some additional tips and tricks to help you factorise algebraic expressions:

Use the distributive property: The distributive property is a powerful tool for factorising expressions. It allows us to expand and simplify expressions by distributing the terms.
Simplify the expression: Once you have identified the common factor, you can simplify the expression by combining the terms.
Check your work: Finally, it is essential to check your work to ensure that the expression has been factorised correctly.
Practice, practice, practice: The more you practice factorising algebraic expressions, the more comfortable you will become with the process.

Real-World Applications

Common Mistakes

Here are some common mistakes to avoid when factorising algebraic expressions:

Not identifying the common factor: Failing to identify the common factor is a common mistake when factorising expressions.
Not using the distributive property: Failing to use the distributive property can lead to incorrect factorisation.
Not simplifying the expression: Failing to simplify the expression can lead to unnecessary complexity.
Not checking your work: Failing to check your work can lead to errors and incorrect factorisation.