Welcome to factorisation! Factorisation is the process of breaking down an algebraic expression into a product of simpler expressions called factors. It's essentially the reverse of expanding or multiplying. For example, we can factor six x squared plus twelve x as six x times the quantity x plus two.
The first and most important method is finding the common factor. Look for the greatest common factor of all terms in the expression, then factor it out. For example, twelve x plus eighteen equals six times the quantity two x plus three. Similarly, four x squared plus eight x equals four x times the quantity x plus two.
For quadratic trinomials, when the coefficient of x squared is one, find two numbers that multiply to the constant term and add to the middle coefficient. For x squared plus five x plus six, we need numbers that multiply to six and add to five, which are two and three. When the leading coefficient is not one, like two x squared plus five x plus three, we can factor it as two x plus three times x plus one.
There are special patterns to recognize. The difference of squares pattern states that a squared minus b squared equals a minus b times a plus b. For example, x squared minus nine equals x minus three times x plus three. For cubes, x cubed plus eight equals x plus two times x squared minus two x plus four.
To summarize factorisation: Always start by looking for common factors. Choose the right method for quadratic expressions. Learn to recognize special patterns like difference of squares and cubes. Always verify your work by expanding back to the original expression. With practice, factorisation becomes much easier and faster.