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Polynomial IdentitiesCovid19 has led the world to go through a phenomenal transition . Elearning is the future today. Stay Home , Stay Safe and keep learning!!! Polynomial Identities : An algebraic expression in which the variables involved have only non negative integral powers is called polynomial.For factorization or for the expansion of polynomial we use the following identities.
Some solved examples : Expand the following using Polynomial Identities. 1) (2a + 5) ^{2} Solution : (2x + 5) ^{2} = (2a) ^{2} + 2(2a)(5) + 5 ^{2} [ using the identity ( x + y ) ^{2} = x ^{2} + 2xy +y ^{2} ] = 4a ^{2} + 20 a + 25 _________________________________________________________________ 2) ( b + 6)(b  6) [ using the identity (x + y)(x – y) = x ^{2 } – y ^{2} ] ( b + 6)(b  6) = b ^{2}  6 ^{2} = b ^{2}  36 __________________________________________________________________ 3) ( 3a  4) ^{3} [using the identity(x  y) ^{3 } = x ^{3}  3x ^{2} y + 3xy ^{2}  y ^{3} ] ( 3a  4) ^{3} = (3a) ^{3}  (3a) ^{2} (4) + 3(3a)(4) ^{2}  4 ^{3} = 27a ^{3} 72a ^{2} + 36a  64 __________________________________________________________________ Factorize the following using Polynomial Identities : 1) Factorize: 64a ^{3}  27b ^{3}  144a ^{2} b + 108ab ^{2} . Solution : 64a ^{3}  27b ^{3}  144a ^{2} b + 108ab ^{2} = (4a) ^{3}  (3b) ^{3}  36ab(4a 3b) = (4a) ^{3}  (3b) ^{3} 3(4a)(3b)(4a 3b) = (4a  3b) ^{3} [ using x ^{3} + y ^{3 } 3xy(x –y) ] = (4a 3b)(4a 3b)(4a 3b) _________________________________________________________________ 2)Evaluate : (104) ^{3} (104) ^{3} = ( 100 + 4) ^{3} = = (100) ^{3} + (4) ^{3} + 3(100)(4)(100 + 4) [Using Identity V] = 1000000 + 64 + 124800 = 1124864 _______________________________________________________________ 3) Evaluate : (–12) ^{3} + (7) ^{3} + (5) ^{3} Solution : (–12) ^{3} + (7) ^{3} + (5) ^{3} From the above we can see that 12 + 7 + 5 = 0 (–12) ^{3} + (7) ^{3} + (5) ^{3} = 3(12)(7)(5) [Using identity 10] = 1260 Polynomial • Degree of the Polynomial • Zeros of Polynomial • Remainder Theorem • Find remainder by Synthetic Division • Rational root test in Polynomial • Solved Examples on Polynomial identities Covid19 has affected physical interactions between people. Don't let it affect your learning.
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