With the term ‘prime factorization’ we can understand that this is the process of factoring a number that will have prime factors; this means the factors of that particular number being factored will be prime numbers. Prime factorization is well explained in this content. This explanation will definitely help the students to easily find out the prime factors of the number.
Prime factorization can be defined as the process of finding the prime factors of a given number. This happens when the original number is factored into other primes only.
Do you know what a composite number is? Well, a composite number is a number that can be factored into two numbers only. This prime factorization will work with these composite numbers.
Examples of prime factorization are as follows:
- Prime factorization of 12 is 2 × 2 × 3 = 22 × 3
- Prime factorization of 18 is 2 × 3 × 3 = 2 × 32
- Prime factorization of 24 is 2 × 2 × 2 × 3 = 23 × 3
- Prime factorization of 20 is 2 × 2 × 5 = 22 × 5
- Prime factorization of 36 is 2 × 2 × 3 × 3 = 2² × 3²
How Does Prime Factorization Help in Finding the HCF and LCM?
Did you know that when a prime number is multiplied with any natural or whole numbers other than 0, then the result is always composite numbers? Thus, we understand that prime factorization is performed on the composite numbers in order to chalk down the prime numbers within that composite number. And this same method is used while calculating the HCF and LCM, that is Highest Common factor and Lowest Common Multiple, respectively.
Do You Know the Algebra Formula?
An algebra is a branch of maths that deals with letters. It uses the letters in place of the numbers. Algebra represents both sides of an equation, one with the numbers and the other with the letters. While both these have the same result and effect.
These numbers are constants. Algebra even includes real numbers, complex numbers, matrices, vectors, and other things. Some of the common letters which represent any kind of algebraic problem and equations are – X, Y, A, and B.
List of Important Formulae in Algebra
Algebra consists of many formulae, so we had made a list for easy studying. This is a list of algebra formula follows:
- a2 – b2 = (a – b)(a + b)
- (a + b)2 = a2 + 2ab + b2
- a2 + b2 = (a + b)2 – 2ab
- (a – b)2 = a2 – 2ab + b2
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
- (a + b)3 = a3 + 3a2b + 3ab2 + b2 ; (a + b)3 = a3 + b3 + 3ab(a + b)
- (a – b)3 = a3 – 3a2b + 3ab2 – b3 = a3 – b3 – 3ab(a – b)
- a3 – b3 = (a – b)(a2 + ab + b2)
- a3 + b3 = (a + b)(a2 – ab + b2)
- (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4
- (a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4
- a4 – b4 = (a – b)(a + b)(a2 + b2)
- a5 – b5 = (a – b)(a4 + a3b + a2b2 + ab3 + b4)
- If n is a natural number an – bn = (a – b)(an-1 + an-2b+…+ bn-2a + bn-1)
- If n is even (n = 2k), an + bn = (a + b)(an-1 – an-2b +…+ bn-2a – bn-1)
- If n is odd (n = 2k + 1), an + bn = (a + b)(an-1 – an-2b +an-3b2…- bn-2a + bn-1)
- (a + b + c + …)2 = a2 + b2 + c2 + … + 2(ab + ac + bc + ….)
- Laws of exponents (am)(an) = am+n ; (ab)m = ambm ; (am)n = amn
From this lesson, we have acquired our knowledge of prime factorization and learned the list of formulae of algebra. The formulae of algebra are of utmost importance; students will resort to them while solving any algebraic equations.
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