Some of zeros of a polynomial with real coefficients are -2 -2- 1-2 - 1-2 -2- - and 1-2- where - is the cube root of unity- Then the minimum degree of the polynomial is

The zeros of a polynomial with real coefficients exist in conjugate pairs. ω+2ω^2ω^2+2ω nbsp; 1+2ω1+2ω^2 nbsp; ωω^2 nbsp; 1 ω ω^2=2→ a real root So there ...

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Standard XII

Mathematics

Quadratic Equation

Some of zeros...

Question

Some of zeros of a polynomial with real coefficients are $ω + 2 ω^{2}, 1 + 2 ω, 1 + 2 ω^{2}, ω$ and $1 - ω - ω^{2},$ where $ω$ is the cube root of unity. Then the minimum degree of the polynomial is

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Solution

The correct option is C $7$
The zeros of a polynomial with real coefficients exist in conjugate pairs.
$ω + 2 ω^{2} conjugate - ---- \to ω^{2} + 2 ω$
$1 + 2 ω conjugate - ---- \to 1 + 2 ω^{2}$

$ω conjugate - ---- \to ω^{2}$
$1 - ω - ω^{2} = 2 \to$ a real root
So there must be at least $7$ roots.
Therefore, minimum degree of the polynomial is $7.$

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Some of zeros of a polynomial with real coefficients are ω+2ω2,1+2ω,1+2ω2,ω and 1−ω−ω2, where ω is the cube root of unity. Then the minimum degree of the polynomial is

Some of zeros of a polynomial with real coefficients are $ω + 2 ω^{2}, 1 + 2 ω, 1 + 2 ω^{2}, ω$ and $1 - ω - ω^{2},$ where $ω$ is the cube root of unity. Then the minimum degree of the polynomial is