Factorisation of Algebraic Expressions

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Factorisation of Algebraic Expressions

Factorisation is a crucial skill in algebra that allows us to simplify and rewrite expressions in more useful forms. In this article, we'll dive into the topic of factorising algebraic expressions, exploring some key concepts and techniques.

Definition and Examples

Factorisation is the process of expressing an expression as the product of two (or more) simpler factors, often involving terms containing common factors. For instance, consider the expression (3x + 6). This expression can be factored as (3(x + 2)).

In general, the process of factorising an expression involves identifying common factors and using them to rewrite the expression as a product of simpler expressions.

Distributive Property

The distributive property plays a fundamental role in factorisation. For example, consider the expression (6x - 3x). We can use the distributive property to rewrite this expression as (3(x - 1)), which is in factored form.

Factoring Linear Expressions

One common type of factorisation involves factoring linear expressions. These are expressions that can be written in the form (ax + b), where (a) and (b) are constants and (x) is a variable.

To factor a linear expression, we look for common factors in its terms and use them to rewrite the expression as a product of simpler factors. For instance, consider the expression (2x - 4). We can factor this expression as (2(x - 2)), which is the product of two simpler expressions.

Factoring Quadratic Expressions

Factoring quadratic expressions, which have the form (ax^2 + bx + c), is a bit more challenging than factoring linear expressions. However, there are some general methods and techniques we can use to accomplish this.

One common method for factoring quadratic expressions involves looking for two binomials that add up to (b) and have a product that equals (ac). For example, consider the expression (x^2 - 5x + 6). We can factor this expression as ((x - 2)(x - 3)), which are two binomials that add up to (-5) and have a product equal to (12).

Factoring by Grouping

In some cases, factoring an expression involves grouping the terms and applying the distributive property. For example, consider the expression (x^2 + 5x + 6). We can rewrite this expression as ((x + 2)(x + 3)) by grouping the first two terms and applying the distributive property to the parentheses.

Practice and Problem Solving

Practicing factorisation is essential for getting comfortable with the process and techniques involved. To help you build your skills, here are a few exercises that you can try:

Factor the following expression: (6x - 9).
Factor the following expression: (2x^2 + 5x + 3).
Factor the following expression: (x^2 - 2x - 15).
Factor the following expression: (8a^2 + 13ab + 6b^2).
Factor the following expression: (3y^2 - 9y + 6).

As you work through these exercises, remember to follow the rules and techniques we've discussed, and don't be afraid to make mistakes. The key is to keep practicing and building your understanding of factorisation. Happy learning!

Factorisation of Algebraic Expressions

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Questions and Answers

Which method is commonly used for factoring quadratic expressions by looking for two binomials that add up to the coefficient of x and have a product equal to the constant term?

What is the correct factorization of the expression: $6x - 9$?

For the expression $2x^2 + 5x + 3$, what is the correct factored form?

Which of the following is the correct factorization of $x^2 - 2x - 15$?

For the expression $8a^2 + 13ab + 6b^2$, what is the correct factored form?

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