If ωis a complex cube root of unity, then (1+5 ω+9 ω^2)/(ω^2+5+9 ω)+(2+3 ω+5 ω^2)/(5+2 ω+3 ω^2)

If ωis a complex cube root of unity, then (1+5 ω+9...

If $\omega$ is a complex cube root of unity, then $\frac{1+5 \omega+9 \omega^{2}}{\omega^{2}+5+9 \omega}+\frac{2+3 \omega+5 \omega^{2}}{5+2 \omega+3 \omega^{2}}$ is equal to
(a) $-1$
(b) 20
(c) 0
(d) $-2 \omega$

Key Concepts

Roots of Unity

Roots of unity are the solutions to the equation z^n = 1 in the complex plane. They form a regular polygon when plotted in the complex plane and are used to simplify and evaluate expressions that involve complex numbers raised to various powers.

Cube Roots of Unity

The cube roots of unity are the solutions to z^3 = 1. They include the real root 1 and two non-real complex roots, usually denoted by ? and ?², which satisfy the relation 1 + ? + ?² = 0. This relation is key for replacing and simplifying expressions containing ? and its powers.

Algebraic Simplification Using Root Identities

By knowing the fundamental identity 1 + ? + ?² = 0 and the minimal polynomial of the cube roots of unity, one can substitute expressions like ?² with -1-?. These substitutions greatly simplify rational expressions and other complex algebraic formulas involving powers of ?.

Transcript

00:01 So in the given question we are told that omega is the complex cube root of unity, that is 1, right? and we are also told, we are told to find if omega is this, we are told to find the value of 1 plus 5 omega plus 9 omega square divided by omega square plus 5.

00:34 5 plus 9 omega plus 2 plus 3 omega plus 3 omega plus 5 omega squared.

00:48 5 plus 2 omega plus 3 omega square.

00:55 So this is the value of the expression.

00:58 The value of this expression is to be found, right? so let's look at this expression.

01:06 What we know is that omega is the cube root, the complex cube root of 1.

01:12 Right.

01:13 So let's start there.

01:15 And take the first part of this expression.

01:20 So we have 1 plus 5 omega plus 9 omega square divided by omega square plus 5 plus 9 omega, right? so since omega is the complex cube root of 1, we can write 1 as omega cube, right? so omega cube plus 5 omega plus 9 omega square is equal to omega square plus 5 plus 9 omega.

01:50 And from the numerator, let's take omega as a common factor.

01:55 Then we would have omega plus omega times omega square plus 5 plus 9 omega which is the same expression as that in the numerator right so now we can cancel off the same term from the numerator and denominator and what we are left with is omega so this is the simplified version of the first term of the expression now let's take the next term which is 2 plus 3 omega plus 5 omega squared divided by 5 plus 2 omega plus 3 omega square.

02:45 And now we can write 2 as 2 times 1 and 1 can be written as omega tube...

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