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.
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.
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.
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.
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:
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.
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).
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:
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.
Factorising algebraic expressions involves the same steps as non-algebraic expressions. Find the GCD and use it to simplify the expression.
If the expression is quadratic, factorise it using the appropriate steps. Otherwise, find the GCD and factorise as usual.
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.
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).