Factorising the Given Expression

Oct 13, 2025 by miodeya.com 41 views

Introduction

Algebraic expressions are a fundamental concept in mathematics, and factorisation is a crucial technique used to simplify and solve equations. In this article, we will focus on factorising a given quadratic expression, which is a polynomial of degree two. The expression we will be working with is: $x^2 - 3xy - 40y^2$ . Our goal is to factorise this expression completely, which means expressing it as a product of simpler expressions.

What is Factorisation?

Factorisation is the process of expressing an algebraic expression as a product of simpler expressions, called factors. These factors can be numbers, variables, or a combination of both. The main purpose of factorisation is to simplify an expression, make it easier to work with, and solve equations.

Types of Factorisation

There are several types of factorisation, including:

Linear factorisation: This involves factoring out a common factor from two or more terms.
Quadratic factorisation: This involves factoring a quadratic expression into the product of two binomial expressions.
Polynomial factorisation: This involves factoring a polynomial expression into the product of simpler polynomial expressions.

Factorising the Given Expression

To factorise the given expression, we need to look for common factors and use the techniques of factorisation. Let's start by factoring out the greatest common factor (GCF) from the expression.

Step 1: Find the GCF

The GCF of the expression $x^2 - 3xy - 40y^2$ is $1$ , since there is no common factor that can be factored out from all the terms.

Step 2: Look for Common Factors

Now, let's look for common factors among the terms. We can see that the terms $x^2$ and $-40y^2$ have a common factor of $-1$ , but this is not a common factor that can be factored out from all the terms.

Step 3: Use the Difference of Squares Formula

The expression $x^2 - 3xy - 40y^2$ can be written as $(x^2 - 40y^2) - 3xy$ . We can see that the first two terms form a difference of squares, which can be factored as $(x - 8y)(x + 5y)$ .

Step 4: Factor the Remaining Term

Now, we have factored the first two terms as $(x - 8y)(x + 5y)$ . The remaining term is $-3xy$ , which can be factored as $-3xy = -3xy$ .

Step 5: Write the Final Factorisation

Combining the results from the previous steps, we can write the final factorisation of the expression as:

x^2 - 3xy - 40y^2 = (x - 8y)(x + 5y) - 3xy

However, we can simplify this further by combining the last two terms:

x^2 - 3xy - 40y^2 = (x - 8y)(x + 5y) - 3xy = (x - 8y)(x + 5y - 3y)

Simplifying further, we get:

x^2 - 3xy - 40y^2 = (x - 8y)(x + 2y)

Therefore, the final factorisation of the expression is:

x^2 - 3xy - 40y^2 = (x - 8y)(x + 2y)

Conclusion

In this article, we factorised the quadratic expression $x^2 - 3xy - 40y^2$ completely. We used the techniques of factorisation, including finding the GCF, looking for common factors, and using the difference of squares formula. The final factorisation of the expression is $(x - 8y)(x + 2y)$ .

Example Problems

Here are some example problems that you can try to practice your factorisation skills:

Factorise the expression $x^2 + 5xy - 6y^2$ .
Factorise the expression $x^2 - 7xy + 12y^2$ .
Factorise the expression $x^2 + 2xy - 15y^2$ .

Solutions

Here are the solutions to the example problems:

Factorise the expression $x^2 + 5xy - 6y^2$ $x^{2} + 5 x y - 6 y^{2}$ :
- Step 1: Find the GCF
- Step 2: Look for common factors
- Step 3: Use the difference of squares formula
- Step 4: Factor the remaining term
- Step 5: Write the final factorisation
- Solution: $(x + 6y)(x - y)$
Factorise the expression $x^2 - 7xy + 12y^2$ $x^{2} - 7 x y + 12 y^{2}$ :
- Step 1: Find the GCF
- Step 2: Look for common factors
- Step 3: Use the difference of squares formula
- Step 4: Factor the remaining term
- Step 5: Write the final factorisation
- Solution: $(x - 4y)(x - 3y)$
Factorise the expression $x^2 + 2xy - 15y^2$ $x^{2} + 2 x y - 15 y^{2}$ :
- Step 1: Find the GCF
- Step 2: Look for common factors
- Step 3: Use the difference of squares formula
- Step 4: Factor the remaining term
- Step 5: Write the final factorisation
- Solution: $(x + 5y)(x - 3y)$

Tips and Tricks

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

Always look for common factors first.
Use the difference of squares formula when possible.
Factor out the GCF from each term.
Use the distributive property to expand and simplify expressions.
Check your work by multiplying the factors together to get the original expression.

Conclusion

Introduction

In our previous article, we discussed the concept of factorisation and how to factorise a quadratic expression. In this article, we will answer some frequently asked questions (FAQs) related to factorisation.

Q: What is factorisation?

A: Factorisation is the process of expressing an algebraic expression as a product of simpler expressions, called factors. These factors can be numbers, variables, or a combination of both.

Q: Why is factorisation important?

A: Factorisation is an important technique used to simplify and solve equations. It helps to make complex expressions easier to work with and understand.

Q: What are the different types of factorisation?

A: There are several types of factorisation, including:

Linear factorisation: This involves factoring out a common factor from two or more terms.
Quadratic factorisation: This involves factoring a quadratic expression into the product of two binomial expressions.
Polynomial factorisation: This involves factoring a polynomial expression into the product of simpler polynomial expressions.

Q: How do I factorise a quadratic expression?

A: To factorise a quadratic expression, follow these steps:

Find the GCF of the expression.
Look for common factors among the terms.
Use the difference of squares formula when possible.
Factor out the GCF from each term.
Use the distributive property to expand and simplify expressions.

Q: What is the difference of squares formula?

A: The difference of squares formula is:

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

This formula can be used to factorise expressions of the form $a^2 - b^2$ .

Q: How do I factorise a polynomial expression?

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

Find the GCF of the expression.
Look for common factors among the terms.
Use the difference of squares formula when possible.
Factor out the GCF from each term.
Use the distributive property to expand and simplify expressions.

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

A: Some common mistakes to avoid when factorising include:

Not finding the GCF of the expression.
Not looking for common factors among the terms.
Not using the difference of squares formula when possible.
Not factoring out the GCF from each term.
Not using the distributive property to expand and simplify expressions.

Q: How do I check my work when factorising?

A: To check your work when factorising, multiply the factors together to get the original expression. If the result is the same as the original expression, then your factorisation is correct.

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

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

Science: Factorisation is used to solve equations in physics and chemistry.
Engineering: Factorisation is used to design and optimize systems.
Computer Science: Factorisation is used in cryptography and coding theory.

Conclusion

Factorisation is a powerful technique used to simplify and solve equations. In this article, we answered some frequently asked questions (FAQs) related to factorisation. We hope that this article has helped to clarify any doubts you may have had about factorisation.