Understanding Factorising Expressions

Have you ever dressed up for a "rag day" event at university and been mistaken for someone else? My friend Biodun and I experienced this as we sported different martial arts costumes. Little did people know, they were witnessing the concept of factorising expressions in action.

But what exactly is factorising expressions? It is the process of simplifying mathematical expressions by finding the greatest common divisor (GCD) outside the bracket and the remaining expression inside. This results in an expanded form that is equivalent to the original. In simpler terms, it's like seeing Biodun and I as "related" due to our similar costumes.

To factorise expressions, we look for similarities or common factors between the terms. For instance, the numbers 6 and 4 both have a common factor of 2, which can divide both numbers without a remainder. This number is the GCD. Similarly, in the numbers 12 and 8, 2 is a common divisor, but 4 gives us the smallest possible results.

The process of factorising an expression involves a few simple steps. First, we may need to rearrange the terms by grouping like terms. Then, we identify the GCD among the terms and divide each term by it. The GCD is placed outside the bracket, while the result (referred to as the "macronym") is placed inside, giving us the desired factorised form.

Let's try out some examples.

• a) What is the GCD between 15x and 25y? The GCD is 5. Dividing each term by 5 gives us the macronyms 3x and 5y. Putting it all together, we have: 5(3x+5y)
• b) In the expression 2x and 4xy, what is the GCD? In this case, the GCD is 2x. Dividing each term by 2x results in the macronyms 1 and 2y, giving us the factorised form: 2x(1+2y)
• c) When dealing with complex expressions like 4ap+2ab-21bq, how do we factorise? In this scenario, we can group the terms 2ab and 4ap together as they have a common factor of 2a. However, in the expression 2ab+4ap-21bq, only b is a common factor between the first two terms. Care must be taken when grouping terms to ensure the correct GCD is identified.

Next, let's move on to factorising and expanding linear expressions. These are expressions where all the variables and constants have an exponent of 1. For example, 3x+4y is a linear expression, while 2x^2+5xy is not as y has an exponent of 2. The steps for factorising remain the same as before.

• a) In the expression 6x+8y, what is the GCD? The GCD is 2. Dividing each term by 2 gives us the macronyms 3x and 4y, resulting in the factorised form: 2(3x+4y)
• b) What is the GCD in 2x+6y+8z? In this case, there is no common factor among all three terms. But, 2 is a common factor between the first two terms, so we factorise them separately and then combine the results, giving us: 2(x+3y+4z)

At first, factorising expressions may seem intimidating. However, with practice, it becomes easier to spot common factors and simplify expressions. Just like how my friend Biodun and I were "factorised" into brothers by our costumes, we can now understand how expressions can be broken down into simpler forms using factorisation.

The Process of Factorising Expressions: Simplifying and Solving

Understanding expressions can be challenging, but with the right techniques, it is possible to simplify and solve them. Let's take a look at some examples and learn how to factorise expressions.

A Step-by-Step Guide for Factoring Expressions

When attempting to factorise an expression, start by expanding it and combining any like terms together. Then, identify a common factor, such as 5 in the expression 5x+5y. Use this factor and follow these steps:

• Divide the expression by the common factor to obtain the GCD outside the brackets
• The result inside the bracket is known as the macronym

Now, let's apply these steps to factorise a quadratic expression. Using the same process, we get:

5(2x+3y)

Remember, if there is a constant multiplying through the expression, it should be brought outside the bracket in the final factorised form.

A Real-Life Example

Imagine Finicky was given 7 oranges and 4 pears, while Indodo received 3 oranges and 11 pears. We can represent Finicky's fruits as 7x+4y and Indodo's fruits as 3x+11y. Their total fruit count would be:

7x+4y + 3x+11y10x+15y

Using the same steps, we can factorise this expression to get the final result of 5(2x+3y).

Simplifying Expressions: Important Steps to Remember

The most important aspect of factorising expressions is finding the greatest common divisor (GCD) to simplify the expression. However, there are a few other crucial steps to consider:

• The GCD is placed outside the brackets, while the result (macronym) is inside
• Linear expressions contain both constants and variables in the first power
• Quadratic expressions follow the same steps, but may require use of the sum and product rule

FAQs About Factoring Expressions

How do you factorise expressions?

To factorise an expression, first identify the GCD and divide it through the expression. The GCD will appear outside the brackets, while the result is inside.

How do you factorise algebraic expressions?

Factorising algebraic expressions involves the same steps as non-algebraic expressions. Find the GCD and use it to simplify the expression.

How do you factorise algebraic expressions with powers?

If the expression is quadratic, factorise it using the appropriate steps. Otherwise, find the GCD and factorise as usual.

Understanding Expression Factoring: Everything You Need to Know

When we talk about factoring an expression, we are referring to the process of simplifying an expression by finding its greatest common divisor (GCD) and dividing it throughout the expression.

An Illustrative Example of Factoring an Expression

To better grasp the concept of factoring an expression, let's look at an example. Consider the expression 6x+3xy. To factor this expression, we first identify the GCD, which in this case is 3x. We then divide this GCD throughout the expression to get the final result of 3x(2+y).