How To Factorise In Maths

Mastering the Art of Factorisation in Maths

Factorisation, a cornerstone of algebra, might seem daunting at first, but with a structured approach and plenty of practice, it becomes a powerful tool for solving mathematical problems. This comprehensive guide will walk you through various factorisation techniques, explaining the underlying principles and providing numerous examples to solidify your understanding. Whether you're a high school student struggling with quadratic equations or an adult brushing up on your math skills, this article will equip you with the knowledge and confidence to tackle factorisation problems effectively. We'll cover common methods like finding common factors, difference of squares, perfect square trinomials, and solving quadratic equations by factoring.

What is Factorisation?

Factorisation, in simple terms, is the process of breaking down a mathematical expression into smaller, simpler expressions that, when multiplied together, give you the original expression. Think of it like reverse multiplication. Just as you can multiply numbers to get a product, factorisation allows you to find the numbers (or expressions) that were multiplied to obtain that product. This process is crucial for simplifying expressions, solving equations, and working with various mathematical concepts.

Common Factor Factorisation: The Building Block

This is the most fundamental technique. It involves identifying common factors within an expression and then factoring them out. A common factor is a term that divides evenly into all terms in the expression.

Example 1: Factorise 6x + 18

Both 6x and 18 are divisible by 6. Therefore, we can factor out 6:

6x + 18 = 6(x + 3)

Example 2: Factorise 4a²b + 8ab² - 12ab

Here, the common factors are 4, a, and b. Factoring these out, we get:

4a²b + 8ab² - 12ab = 4ab(a + 2b - 3)

Example 3: Factorise 3x(x+2) + 6(x+2)

Notice that (x+2) is a common factor here:

3x(x+2) + 6(x+2) = (x+2)(3x+6) = 3(x+2)(x+2) = 3(x+2)²

This example highlights that sometimes, you might need to factor out common factors more than once to fully factorise an expression.

Difference of Squares: A Special Case

The difference of squares is a specific type of expression that can be easily factorised. It follows the pattern: a² - b² = (a + b)(a - b)

Example 4: Factorise x² - 25

Here, a = x and b = 5 (because 25 = 5²). Therefore:

x² - 25 = (x + 5)(x - 5)

Example 5: Factorise 4x² - 9y²

Here, a = 2x and b = 3y:

4x² - 9y² = (2x + 3y)(2x - 3y)

Perfect Square Trinomials: Recognising the Pattern

A perfect square trinomial is a trinomial (an expression with three terms) that can be factored into the square of a binomial. It follows the pattern: a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)²

Example 6: Factorise x² + 6x + 9

Here, a = x and b = 3 (because 9 = 3² and 6x = 2 * x * 3). Therefore:

x² + 6x + 9 = (x + 3)²

Example 7: Factorise 4x² - 12x + 9

Here, a = 2x and b = 3:

4x² - 12x + 9 = (2x - 3)²

Factorising Quadratic Expressions: The General Case

Quadratic expressions are of the form ax² + bx + c, where a, b, and c are constants. Factorising these can be more challenging, but there are several methods:

• Method 1: Finding factors of 'ac' that add up to 'b'

This method involves finding two numbers that multiply to 'ac' and add up to 'b'. Let's illustrate:

Example 8: Factorise 2x² + 7x + 3

Here, a = 2, b = 7, and c = 3. 'ac' = 6. We need two numbers that multiply to 6 and add up to 7. These numbers are 6 and 1. We then rewrite the expression:

2x² + 7x + 3 = 2x² + 6x + x + 3

Now, factor by grouping:

= 2x(x + 3) + 1(x + 3)

= (x + 3)(2x + 1)

• Method 2: Using the Quadratic Formula

If the above method proves difficult, the quadratic formula can always be used to find the roots (solutions) of the quadratic equation ax² + bx + c = 0. The roots are given by:

x = (-b ± √(b² - 4ac)) / 2a

Once you have the roots, say α and β, the factorised form is a(x - α)(x - β).

Example 9: Factorise 3x² - 5x - 2 using the quadratic formula.

a = 3, b = -5, c = -2. Using the quadratic formula:

x = (5 ± √(25 - 4 * 3 * -2)) / 6 = (5 ± √49) / 6 = (5 ± 7) / 6

Therefore, the roots are x = 2 and x = -1/3. The factorised form is:

3(x - 2)(x + 1/3) = (x - 2)(3x + 1)

Factorisation of Cubic and Higher-Degree Polynomials

Factorising cubic and higher-degree polynomials is significantly more complex. Methods include:

• Factoring out common factors: Always start by checking for common factors among all terms.
• Using the Rational Root Theorem: This theorem helps identify potential rational roots (roots that are fractions).
• Synthetic Division: This is a method for efficiently dividing a polynomial by a linear factor (x - r), where 'r' is a root.
• Grouping: Similar to factorising quadratic equations, grouping terms can sometimes reveal common factors.

Solving Equations by Factorisation

Factorisation is a powerful tool for solving equations, particularly quadratic equations. If you can factor an equation into the form (x - a)(x - b) = 0, then the solutions are x = a and x = b. This is because the product of two factors is zero only if at least one of the factors is zero.

Example 10: Solve the equation x² - 5x + 6 = 0

Factorising the quadratic gives: (x - 2)(x - 3) = 0

Therefore, the solutions are x = 2 and x = 3.

Frequently Asked Questions (FAQ)

Q1: What if I can't find the factors easily?

A1: If you're struggling to find factors directly, consider using the quadratic formula for quadratic expressions or exploring more advanced techniques for higher-degree polynomials. Practice is key – the more you work through examples, the better you'll become at recognizing patterns and finding factors.

Q2: Is there a specific order I should follow when factorising?

A2: Yes, a systematic approach is helpful. Start by checking for common factors. Then, look for special patterns like difference of squares or perfect square trinomials. For quadratic expressions, try factoring using the methods described above. Finally, for higher-degree polynomials, explore techniques like the Rational Root Theorem and synthetic division.

Q3: Why is factorisation important?

A3: Factorisation is a fundamental skill in algebra with numerous applications. It simplifies expressions, helps solve equations, and is used in many advanced mathematical concepts, including calculus and differential equations.

Conclusion: Embrace the Challenge of Factorisation

Factorisation, while initially challenging, is a skill that becomes second nature with consistent practice. Mastering these techniques will not only improve your algebraic skills but also provide a solid foundation for more advanced mathematical concepts. Remember to approach each problem systematically, starting with the simplest methods and progressing to more complex ones as needed. Don't be discouraged by difficulties; persistence and practice are the keys to success in mastering the art of factorisation. With dedication, you can unlock its power and confidently tackle a wide range of mathematical problems.