Question

Consider the polynomial identity: (x+y) ^3 = x^3 +3x^2y + 3xy^2 +y ^3

Answer 1

Step-by-step explanation:

Polynomial identity: $(x+y)^{3} = x^{3} + y^{3} + 3x^{2}y + 3xy^{2}$

(A) Expanding above identity by LHS:

$(x+y)^{3}\\(x+y)(x+y)(x+y)\\(x+y)(x^{2}+2xy+y^{2})\\(x^{3}+2x^{2}y+y^{2}x+x^{2}y+2xy^{2}+y^{3})\\(x^{3}+3y^{2}x+3xy^{2}+y^{3})$ Hence , proved.

(B) Value of ,  $11^{3} = 11(11)(11) = 121(11) = 1331$. Finding the same with help of equation:

$11^{3} = (10+1)^{3}\\11^{3} = 10^{3} + 3(1^{2})10+3(1)(10^{2})+1^{3}\\11^{3} = 1000 + 30 + 300 + 1\\11^{3} = 1331$ Hence , Both produce same result.