Factorization in Mathematics
Introduction to Factorization
Factorization is the process of breaking down an expression into factors that, when multiplied together, give the original
expression. In mathematics, factorization is commonly used to simplify expressions, solve equations, and find the roots or
zeros of polynomials.
Types of Factorization
There are several common types of factorization techniques, including:
1. Factorization of Quadratic Polynomials
2. Difference of Squares
3. Factorization by Grouping
4. Sum and Difference of Cubes
5. Prime Factorization
Factorization of Quadratic Polynomials
A quadratic polynomial is in the form ax² + bx + c. The goal of factorizing a quadratic expression is to express it as a product
of two binomials: (px + q)(rx + s).
To factor a quadratic polynomial, look for two numbers that multiply to give ac (the product of a and c) and add up to give b
(the middle term). For example, for the quadratic expression x² + 5x + 6, find two numbers that multiply to 6 (ac) and add to
5 (b), which are 2 and 3. Therefore, the factorization is:
(x + 2)(x + 3)
Difference of Squares
The difference of squares formula states that:
(a² - b²) = (a + b)(a - b)
This is a useful identity for factorizing expressions in the form of a² - b².
Example: Factor x² - 9.
This can be written as (x + 3)(x - 3), as 9 is a perfect square (3²).
Factorization by Grouping
Factorization by grouping is often used when a polynomial has four terms. The idea is to group the terms in pairs and factor
each pair separately.
Example: Factor x² + 5x + 2x + 10.
Group the terms as (x² + 5x) and (2x + 10), then factor each group:
(x(x + 5)) + 2(x + 5)
Now factor out the common binomial factor:
(x + 5)(x + 2)
Sum and Difference of Cubes
The sum and difference of cubes are factorized using the following formulas:
Sum of cubes: a³ + b³ = (a + b)(a² - ab + b²)
Difference of cubes: a³ - b³ = (a - b)(a² + ab + b²)
Example: Factor x³ - 8.
This is a difference of cubes, as 8 = 2³. Apply the formula:
(x - 2)(x² + 2x + 4)
Prime Factorization
Prime factorization is the process of breaking down a number into its prime factors. Every positive integer greater than 1 can
be written as a product of prime numbers.
Example: Prime factorization of 36:
36 = 2 × 2 × 3 × 3, or 36 = 2² × 3².
Summary of Factorization
Factorization is a key mathematical technique used to simplify expressions, solve equations, and understand mathematical
relationships. By recognizing common patterns and applying the appropriate formulas, we can easily break down complex
expressions into simpler factors.
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