Factorising is an essential skill in mathematics that allows you to simplify algebraic expressions, solve equations, and find the roots of quadratic equations. But what exactly is factorising in maths? And how can you factorise easily?
Factorising is the process of writing an algebraic expression as a product of factors. It involves finding two numbers that multiply together to give the constant term of the expression and add up to the coefficient of the middle term. This technique can be applied to various types of expressions, including quadratics, trinomials, and expressions with common factors.
By learning the techniques and methods for factorising, you can easily simplify and solve algebraic equations and expressions. In this article, we will explore what factorising is, how to factorise algebraic expressions and equations, and various factorisation methods. Whether you’re studying algebra or preparing for exams, mastering the art of factorising will pave the way for your success in mathematics.
What is Factorising in Maths?
Factorising in maths refers to the process of breaking down an algebraic expression into its factors. It is the reverse process of expanding (multiplying out) brackets. By factorising an expression, you can simplify it and identify any common factors that may be present. This allows you to simplify algebraic expressions, solve equations, and find the roots of quadratic equations. Factorising expressions is an important skill in algebra that helps in solving complex problems and simplifying calculations.
How to Factorise Quadratics?
To factorise quadratics, you can use different methods to break down the quadratic expression into its factors. By finding the correct factors, you can simplify the expression, solve quadratic equations, and determine the roots of the equation. Let’s explore two common factorisation methods: trial and error and the quadratic formula.
Trial and Error Method
One way to factorise quadratics is by using the trial and error method. This involves finding two numbers that multiply to give the constant term of the quadratic and add up to the coefficient of the middle term. Here’s an example:
Consider the quadratic expression: x^2 + 7x + 10
| Step | Explanation |
|---|---|
| 1 | Determine the constant term and the coefficient of the middle term. |
| 2 | Identify two numbers that multiply to give the constant term (10) and add up to the coefficient of the middle term (7). |
| 3 | In this case, the numbers are 2 and 5, as 2 × 5 = 10 and 2 + 5 = 7. |
| 4 | Rewrite the quadratic expression as a product of two binomial factors using the numbers found in the previous step. |
| 5 | The factored form of the quadratic expression is: (x + 2)(x + 5) |
The quadratic expression x^2 + 7x + 10 has been factorised to (x + 2)(x + 5) using the trial and error method.
The Quadratic Formula
Another method to factorise quadratics is by using the quadratic formula. The quadratic formula provides the roots of a quadratic equation, which can be used to factorise the quadratic expression. The formula is:
x = (-b ± √(b^2 – 4ac)) / 2a
Where a, b, and c are the coefficients of the quadratic equation: ax^2 + bx + c = 0.
Let’s factorise the quadratic expression: x^2 + 5x + 6 using the quadratic formula.
| Step | Explanation |
|---|---|
| 1 | Determine the values of a, b, and c. |
| 2 | Substitute the values into the quadratic formula. |
| 3 | Solve for x using the quadratic formula. |
| 4 | The roots of the quadratic equation are x = -2 and x = -3. |
| 5 | Rewrite the quadratic expression as a product of two binomial factors using the roots found in the previous step. |
| 6 | The factored form of the quadratic expression is: (x + 2)(x + 3) |
The quadratic expression x^2 + 5x + 6 has been factorised to (x + 2)(x + 3) using the quadratic formula.
By applying these factorisation methods, you can easily factorise quadratics, solve quadratic equations, and simplify algebraic expressions.
How to Factorise Equations?
Factorising equations is a valuable technique that simplifies complex equations and aids in finding their solutions. By breaking down an equation into its factors, you can identify common factors and apply specific factoring methods. Let’s explore some key methods for factorising equations:
Factorising Polynomials
Polynomial equations can be factorised using various methods, such as synthetic division or algebraic long division. These methods help in breaking down the polynomial into its irreducible factors, allowing for easier manipulation and solution.
Factor Trees
Factor trees are a visual representation of how an equation’s factors can be identified. By listing the prime factors of each term in the equation, you can construct a factor tree that reveals the common factors and simplifies the equation.
Factorisation Methods
Factorisation methods provide systematic approaches to factorising equations based on their specific characteristics. For example, trinomial equations can be factorised by splitting the middle term into two terms that have common factors with the first and last terms. This method simplifies the equation and facilitates the identification of its factors.
By mastering these factorisation techniques and methods, you can effectively factorise equations, simplify complex expressions, and find solutions effortlessly.
Next, we will delve into the concept of fully factorising expressions and its significance in algebraic calculations and problem-solving.
How to Fully Factorise?
Fully factorising an expression is the process of expressing it as a product of its irreducible factors, including both linear and quadratic factors. This technique is especially useful when dealing with cubic equations or when finding the prime factors of a number. To fully factorise an expression, follow these steps:
- Step 1: Factor out any common factors
- Step 2: Factorise any remaining quadratics and trinomials
- Step 3: Apply prime factorisation to any remaining terms
Let’s take a closer look at each step.
Step 1: Factor out any common factors
| Expression | Factorised Form |
|---|---|
| 2x + 4 | 2(x + 2) |
| 3y^2 + 9y | 3y(y + 3) |
Step 2: Factorise any remaining quadratics and trinomials
| Expression | Factorised Form |
|---|---|
| x^2 + 5x + 6 | (x + 2)(x + 3) |
| 2x^2 + 9x + 5 | (2x + 1)(x + 5) |
Step 3: Apply prime factorisation to any remaining terms
| Expression | Factorised Form |
|---|---|
| 2x^2 + 3x + 5 | Prime factors cannot be further simplified |
| 4x^3 + 8x^2 + 16x | 4x(x + 2)(x + 2) |
Fully factorising an expression helps in simplifying it as much as possible and identifying all the factors that make up the expression. This process is essential for solving equations, simplifying algebraic expressions, and finding the prime factors of numbers.
How to Factorise Algebraic Expressions with Powers?
Factorising algebraic expressions with powers is an important skill in mathematics. By breaking down these expressions into their factors, you can simplify complex equations and solve for variables. The same principles of factoring apply to expressions with powers as with other types of expressions.
Here are the steps to factorise algebraic expressions with powers:
- Identify any common factors: Look for factors that are common to all terms in the expression. This can help simplify the expression before further factoring.
- Break down the expression into its factors: Use methods like trial and error or the quadratic formula to factorise the expression. For expressions with powers, you may need to factorise individual terms within the expression.
- Simplify the expression: Once you have factored the expression, simplify it as much as possible. Combine like terms and rearrange the factors to achieve the simplest form.
Factorising algebraic expressions with powers can be best understood through examples. Let’s consider the following expression:
Example:
Factorise the expression: 3x2 + 6x + 3
To factorise this expression, we first look for any common factors. In this case, all the terms have a common factor of 3:
| Expression | Common Factor |
|---|---|
| 3x2 | 3 |
| 6x | 3 |
| 3 | 3 |
Factoring out the common factor of 3, we get:
3(x2 + 2x + 1)
Next, we need to factorise the quadratic expression within the parentheses. The quadratic expression factors into (x + 1)(x + 1) or (x + 1)2:
3(x + 1)2
So, the fully factorised form of the expression is 3(x + 1)2.
By factorising algebraic expressions with powers, we can simplify complex expressions, solve equations, and find the roots or solutions to the equations.
Conclusion
Mastering the skill of factorising is essential for simplifying algebraic expressions, solving equations, and finding the roots of quadratic equations. By learning and applying various factorisation methods, such as factoring out common factors, factorising quadratics and trinomials, and using prime factorisation, you can confidently approach complex mathematical problems.
Practice is key to becoming proficient in factorising. By consistently working through different types of factorisation examples and familiarising yourself with the techniques involved, you can enhance your problem-solving skills and become more efficient in dealing with algebraic calculations and equations. Factorising expressions will no longer be a daunting task, but rather an empowering tool for simplification and solution-seeking.
Remember, factorisation not only simplifies mathematical expressions but also enables you to grasp the underlying structure and relationships within the equations. By breaking down an expression into its factors, you gain a deeper understanding of its components and uncover valuable insights. So, keep exploring and refining your factorisation skills to unlock new possibilities and conquer the world of mathematics.