Factorisation is the process of breaking down numbers or expressions into their building blocks, called factors. Think of it like taking apart a LEGO structure to see what pieces it's made from. For example, the number 12 can be broken down into 3 times 4. Here we can see 12 blocks arranged in a 3 by 4 grid, showing us that 3 and 4 are factors of 12. Factorisation is simply the reverse of multiplication - instead of multiplying factors together, we're finding what factors multiply to give us our original number.
To find common factors in algebraic expressions, we follow a systematic approach. First, we identify the highest common factor of the coefficients. Let's look at the expression 6x plus 9. We create factor trees for 6 and 9. Six factors into 2 times 3, and 9 factors into 3 times 3. The highest common factor is 3. Next, we factor out this common term: 6x equals 3 times 2x, and 9 equals 3 times 3. Therefore, 6x plus 9 equals 3 times the quantity 2x plus 3. This process helps us simplify expressions and solve equations more easily.
When factoring algebraic expressions with variables, we extend our previous method to include both numerical and variable factors. Let's look at 4x plus 8y. We break down each term: 4x equals 4 times x, and 8y equals 4 times 2y. The common factor is 4, so we get 4 times the quantity x plus 2y. For our second example, 3a squared plus 6a, we have 3a squared equals 3a times a, and 6a equals 3a times 2. Here the common factor is 3a, giving us 3a times the quantity a plus 2. Notice how we look for the highest common factor that includes both numbers and variables.
Factoring by grouping is used for expressions with four terms. Let's work with ax plus bx plus ay plus by. First, we group terms strategically in pairs: the first group is ax plus bx, shown in blue, and the second group is ay plus by, shown in red. Next, we factor each group separately. From the first group, we factor out x to get x times the quantity a plus b. From the second group, we factor out y to get y times the quantity a plus b. Notice both groups now contain the common binomial factor a plus b. Finally, we factor out this common binomial to get the quantity a plus b times the quantity x plus y. This method transforms a four-term expression into a product of two binomials.
Factorisation has many practical applications in mathematics. When simplifying algebraic fractions like 6x plus 9 over 3x plus 12, we first factor both numerator and denominator. The numerator becomes 3 times 2x plus 3, and the denominator becomes 3 times x plus 4. We can cancel the common factor of 3, leaving us with 2x plus 3 over x plus 4. For solving equations like x squared plus 5x equals zero, we factor out x to get x times x plus 5 equals zero, giving us solutions x equals zero or x equals negative 5. When finding rectangle dimensions from an area expression like 2x squared plus 6x, we factor to get 2x times x plus 3, revealing the width as 2x and length as x plus 3. Remember: always look for common factors first, choose the appropriate method, and check your answer by expanding.