Factorising
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What is Factorizing?
Factorizing is the process of breaking down an algebraic expression into a product of simpler expressions, called factors. This technique is fundamental in algebra, as it simplifies solving equations and simplifying expressions.
Basic Structure
An algebraic expression that can be factorized has the form:
Ax2 + Bx + C
Where A, B, and C are constants, and x is a variable. The goal is to rewrite this expression as a product of two binomials.
Steps to Factorize Quadratic Expressions
1. Finding Common Factors
The first step in factorizing any algebraic expression is to look for common factors in all terms. If all terms share a common factor, factor it out.
Example:
4x2 + 8x
Common factor: 4x
Factor out the common factor: 4x(x + 2)
2. Factoring Quadratics
When no common factors exist, and you have a quadratic expression of the form Ax2 + Bx + C, follow these steps:
- Find two numbers that multiply to
A × Cand add toB. - Rewrite the middle term using these two numbers.
- Factor by grouping.
Example:
x2 + 5x + 6
Numbers that multiply to 1 × 6 and add to 5 are 2 and 3.
Rewrite the expression: x2 + 2x + 3x + 6
Factor by grouping: x(x + 2) + 3(x + 2)
Result: (x + 3)(x + 2)
3. Special Cases
Some quadratic expressions follow special patterns that make them easier to factor:
- Difference of Squares:
a2 - b2 = (a - b)(a + b) - Perfect Square Trinomials:
a2 ± 2ab + b2 = (a ± b)2
Examples:
1. x2 - 9
Difference of squares: (x - 3)(x + 3)
2. x2 + 6x + 9
Perfect square trinomial: (x + 3)2
Key Points to Remember
- Always look for common factors first.
- Identify special patterns like the difference of squares or perfect square trinomials for quicker factorization.
- Practice the process of finding numbers that multiply to
A × Cand add toBfor quadratic expressions. - When stuck, verify your factors by expanding them to ensure they multiply back to the original expression.