If x=a+b, y=a ω+b ω^2 and z=a ω^2+b ω, the value of x^3+y^3+z^3 is equal to (a) a^3+b^3 (b) 0 (c

If x=a+b, y=a ω+b ω^2 and z=a ω^2+b ω, the value of...

Key Concepts

Cube Roots of Unity

Cube roots of unity are the three complex solutions to the equation x³ = 1. They consist of the real number 1 and two complex numbers, commonly denoted as ? and ?², which satisfy ?³ = 1 and the relation 1 + ? + ?² = 0. These properties allow for significant simplifications in expressions involving sums and powers when the roots are used as coefficients.

Symmetric Functions and Polynomial Identities

Symmetric functions are expressions that remain unchanged under any permutation of their variables. In polynomial expressions involving sums of cubes or other powers, classical identities such as the sum of cubes formula can help in transforming and simplifying the expressions. This concept is crucial in problems where variables are defined in a symmetric manner, allowing the use of invariance properties to reduce the problem to simpler components.

Linear Transformations Involving Roots of Unity

Linear transformations that incorporate roots of unity serve to change the basis of the variables, often simplifying the analysis of the expression by exploiting the orthogonality and cyclic symmetry of the roots. This approach is particularly effective when variables are mixed through coefficients that are roots of unity, as it can lead to decoupling of the components and reveal hidden symmetric structures in the problem.

Transcript

00:01 So in the given question you are told that if x is equal to a plus b and y is equal to a omega plus b and y is equal to a omega plus b omega square and z is equal to a omega square plus b omega then we need to find the value of x cube plus y cube plus z cube right so over here omega is the complex cube root of 1 right it is the complex cube root of 1 and what we're going to do first is to take the sum of x plus y plus z and check what the sum is going to be so we will have a plus b plus a omega square plus b omega square plus a omega square plus b omega and we should know a property of omega which is a plus one plus omega plus omega square gives us the value as zero so keeping this in mind let's rearrange this expression over here as a plus a omega plus a omega square and b plus b plus b omega plus b omega square and we can take a and b as common factors from this.

01:35 So, a times 1 plus omega plus omega square plus b times 1 plus omega plus omega plus omega square.

01:46 So since 1 plus omega plus omega square is equal to 0, we will end up with a times 0 plus b times 0, which is equal to 0, right? so since we got x plus y plus z is equal to zero, there is a relation that says if x plus y plus z is equal to zero, the value of x cubed plus y cube plus z cube is equal to 3x y z.

02:18 So let's use this relation in order to find the value of x cube plus y cube plus x cube plus x cube plus right.

02:25 So what we will have is 3 times a plus b plus 3 times a omega plus b sorry not plus times a omega plus b omega square which is y and a omega square plus b omega which is z right so x y and z now let's start by expanding these two terms right so what we will have is three times a plus b times a square omega cube plus a b omega cube plus a b omega raised to 4 plus ab omega square plus b b omega squared so since omega is omega is the cube root of 1 omega cube or omega raised to any power that is a multiple of three gives us the value one and omega raised to 4 can be written as omega cube times omega which we can write since omega cube is equal to 1 we can write it as omega right so let's make these substitutions in the given expression so what we will have is 3 times a plus b times a square plus ab omega plus ab omega plus ab omega square plus b square right and now we have ab omega plus ab omega square so we have we know the relation that 1 plus omega plus omega square is equal to 0 from this relation we can take omega plus omega square would then be equal to negative 1 right so here when we take ab as a common factor, what we will have is ab times what we will have this a square plus ab times omega plus omega square plus b square and omega plus omega square is equal to minus 1.

04:53 So it is 3 times a plus b times a square minus ab plus b square...

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