Factorization, also known as factorizing or factoring, is the process of breaking down a mathematical expression or number into its constituent factors. In mathematics, factors are numbers or algebraic expressions that, when multiplied together, produce the original expression or number.
Factorization is commonly used in various areas of mathematics, including algebra, number theory, and calculus. It plays a crucial role in solving equations, simplifying expressions, finding the roots of polynomials, and understanding the properties of numbers.
Factorization of numbers involves breaking down a given number into its prime factors, which are prime numbers that can’t be further divided.
Factorization of 24
Step 1: Start with the smallest prime factor, which is 2. Divide 24 by 2 to get 12.
Step 2: Continue dividing 12 by 2 to get 6, and then divide 6 by 2 to get 3.
Step 3: 3 is a prime number. The prime factors of 24 are 2, 2, 2, and 3.
So, the factorization of 24 is 2 × 2 × 2 × 3, or simply 2^3 × 3.
Factorization of 45
Step 1: Start with the smallest prime factor, which is 3. Divide 45 by 3 to get 15.
Step 2: 15 is not a prime number. Continue with the next smallest prime factor, which is 5. Divide 15 by 5 to get 3.
Step 3: 3 is a prime number. The prime factors of 45 are 3 and 5.
So, the factorization of 45 is 3 × 5.
Factorization of algebraic expressions involves expressing an expression as a product of its factors. The goal is to find common factors or use suitable methods like the difference of squares or perfect square trinomials to factorize.
Factorization of x2 – 4
This expression is a difference of squares, which can be factored using the formula: a2 – b2 = (a + b)(a – b).
So, x2 – 4 can be factorized as (x + 2) (x – 2).
Factorization of 9×2 + 6x
First, find the common factor in the expression, which is 3x.
So, 9×2 + 6x can be factorized as 3x(3x + 2).
Prime factorization is also used to find the least common multiple (LCM) and greatest common divisor (GCD) of numbers.
LCM of 12 and 18
Find the prime factors of 12: 22 × 3
Find the prime factors of 18: 2 × 32
LCM is the product of all unique prime factors raised to their highest power: LCM (12, 18) = 22 × 32 = 36.
GCD of 24 and 36
Find the prime factors of 24: 23 × 3
Find the prime factors of 36: 22 × 32
GCD is the product of all common prime factors raised to their lowest power: GCD (24, 36) = 22 × 3 = 12.