what is factorization explain
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Factorisation :
Factorization is the decomposition of an object (for example, a number, a polynomial) into a product of other objects, or factors, which when multiplied together give the original one.
Example :
For factoring n = 1386 into primes:
1) Start with division by 2: the number is even, and n = 2 · 693. Continue with 693, and 2 as a first divisor candidate.
2) 693 is odd (2 is not a divisor), but is a multiple of 3: one has 693 = 3 · 231 and n = 2 · 3 · 231. Continue with 231, and 3 as a first divisor candidate.
3) 231 is also a multiple of 3: one has 231 = 3 · 77, and thus n = 2 · 32 · 77. Continue with 77, and 3 as a first divisor candidate.
4) 77 is not a multiple of 3, since the sum of its digits is 14, not a multiple of 3. It is also not a multiple of 5 because its last digit is 7. The next odd divisor to be tested is 7. One has 77 = 7 · 11, and thus n = 2 · 32 · 7 · 11. This shows that 7 is prime (easy to test directly). Continue with 11, and 7 as a first divisor candidate.
5) As 72 > 11, one has finished. Thus 11 is prime, and the prime factorization is
Summary :1386 = 2 · 32 · 7 · 11.
Example :
From polynomial
1) x²+4x+4
x²+2x+2x+4
x(x+2)+2(x+2)
(x+2)(x+2)
x=-2 and x=-2 is a zero
and its factors is (x+2) and (x+2)
Hence proved
I think it is helpful
Factorization is the decomposition of an object (for example, a number, a polynomial) into a product of other objects, or factors, which when multiplied together give the original one.
Example :
For factoring n = 1386 into primes:
1) Start with division by 2: the number is even, and n = 2 · 693. Continue with 693, and 2 as a first divisor candidate.
2) 693 is odd (2 is not a divisor), but is a multiple of 3: one has 693 = 3 · 231 and n = 2 · 3 · 231. Continue with 231, and 3 as a first divisor candidate.
3) 231 is also a multiple of 3: one has 231 = 3 · 77, and thus n = 2 · 32 · 77. Continue with 77, and 3 as a first divisor candidate.
4) 77 is not a multiple of 3, since the sum of its digits is 14, not a multiple of 3. It is also not a multiple of 5 because its last digit is 7. The next odd divisor to be tested is 7. One has 77 = 7 · 11, and thus n = 2 · 32 · 7 · 11. This shows that 7 is prime (easy to test directly). Continue with 11, and 7 as a first divisor candidate.
5) As 72 > 11, one has finished. Thus 11 is prime, and the prime factorization is
Summary :1386 = 2 · 32 · 7 · 11.
Example :
From polynomial
1) x²+4x+4
x²+2x+2x+4
x(x+2)+2(x+2)
(x+2)(x+2)
x=-2 and x=-2 is a zero
and its factors is (x+2) and (x+2)
Hence proved
I think it is helpful
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