What is factorization?
Answer:
Factorization is a fundamental concept in algebra where we express a mathematical expression as a product of its factors. This is especially useful in simplifying expressions and solving equations. We’ll explore the topic by breaking down the process into steps and addressing different methods used in factorization.

Basic Factorization:
 To begin with, we factorize simple algebraic expressions such as monomials and binomials.
 For example, consider the expression 6x^2 + 9x. The common factor here is 3x, so we can factorize it as:6x^2 + 9x = 3x(2x + 3)

Factorizing Quadratic Expressions:
 A quadratic expression has the form ax^2 + bx + c. The factorization aims to express it as a product of two binomials.
 For example, for x^2 + 5x + 6, we find two numbers that multiply to 6 (the constant term) and add up to 5 (the coefficient of x). In this case, 2 and 3 are the numbers:x^2 + 5x + 6 = (x + 2)(x + 3)

Difference of Squares:
 The difference of squares is a special factorization pattern where a^2  b^2 can be written as (a + b)(a  b).
 For example, for the expression x^2  9, we can rewrite 9 as 3^2 to apply the difference of squares:x^2  9 = (x + 3)(x  3)

Completing the Square:
 This method involves turning a quadratic expression into a perfect square trinomial. It’s often used to solve quadratic equations.
 For example, for x^2 + 4x + 4, we can write it as:x^2 + 4x + 4 = (x + 2)^2

Factorization by Grouping:
 This method applies to polynomials with four or more terms. We group terms in pairs and factor out the common factors in each group.
 For example, consider x^3 + 3x^2 + x + 3. We can group as follows and factorize each group:x^3 + 3x^2 + x + 3 = x^2(x + 3) + 1(x + 3) = (x^2 + 1)(x + 3)
Final Answer:
Factorization involves expressing mathematical expressions as products of their factors, which simplifies solving and manipulating algebraic expressions. Techniques include factorizing simple expressions, quadratics, recognizing patterns like difference of squares, completing the square, and grouping terms in polynomials. Following these methods allows for effective factorization of various types of algebraic expressions.