Complex Numbers 1+z+z2+z3+....zn-1=(z-a1)(z-a2)(z-an-1) HOW??

Since 1, a1 , a2 , ....... are the roots of zn-1=0 thereforezn-1 = (z-1)(z-a1)(z-a2)(z-a3).......(z - an-1)=> (zn-1)/(z-1) = (z-a1)(z-a2)(z-a3)....(z - an-1) => 1+z+z2+z3+...... zn-1 = (z-a1)(z-a2)(z-a3)....(z - an-1) [ since (zn-1)/(z-1) is the sum of Geometric Progression with 1st term 1 and common ratio z]Hence Proved

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Grade 12Algebra

Complex Numbers

1+z+z²+z³+....z^n-1=(z-a₁)(z-a₂)(z-a_n-1)

HOW??

Kaushik Ray

10 Years agoGrade 12

1 Answer

Jayant Kishore

10 Years ago

Since 1, a₁ , a_{2 ,}....... are the roots of zⁿ-1=0 therefore

zⁿ-1 = (z-1)(z-a₁)(z-a₂)(z-a₃).......(z - a_n-1)

=> (zⁿ-1)/(z-1) = (z-a₁)(z-a₂)(z-a₃)....(z - a_n-1)

=> 1+z+z²+z³+...... z^{n-1 =} (z-a₁)(z-a₂)(z-a₃)....(z - a_n-1) [ since (zⁿ-1)/(z-1) is the sum of Geometric Progression with 1^st term 1 and common ratio z]

Hence Proved