Question

if alpha and beta are the zeroes of n^2 - n - 2 = 0 find another polynomial whose zeroes are 2 alpha and 2 beta​

Answer 1

$\Large{\underline{\underline{\mathfrak{\bf{Question}}}}}$

if α and β are the zeroes of n² - n - 2 = 0 find another polynomial whose zeroes are

2α and 2β .

$\Large{\underline{\underline{\mathfrak{\bf{Solution}}}}}$

$\Large{\underline{\mathfrak{\bf{Given}}}}$

• Equation, n² - n -2 = 0 .........(1)
• α and β are zeroes .

$\Large{\underline{\mathfrak{\bf{Given}}}}$

• Polynomial whose zeroes are 2α and 2β

$\Large{\underline{\underline{\mathfrak{\bf{Explanation}}}}}$

we know,

$\small\boxed{\sf{\:sum\:of\:zeroes\:=\:\dfrac{-(coefficient\:of\:n)}{(coefficient\:of\:n^2)}}}$

$\mapsto\sf{\:(\alpha + \beta)\:=\:\dfrac{-(-1)}{1}} \\ \\ \mapsto\sf{\:(\alpha + \beta)\:=\:1.......(1)}$

Again,

$\small\boxed{\sf{\:product\:of\:zeroes\:=\:\dfrac{(constant\:part)}{(coefficient\:of\:n^2)}}}$

$\mapsto\sf{\:\alpha.\beta\:=\:\dfrac{(-2)}{1}} \\ \\ \mapsto\sf{\:\alpha.\beta\:=\:-2......(2)}$

we have,

$\small\boxed{\sf{\:(\alpha - \beta)\:=\:\sqrt{(\alpha + \beta)^2-4.\alpha.\beta}}} \\ \\ \small\sf{\:\:\:\:keep\:value\:by\:equ(1)\:and\:equ(2)} \\ \\ \mapsto\sf{\:(\alpha-\beta)\:=\:\sqrt{(1)^2-4.(-2)}} \\ \\ \mapsto\sf{\:(\alpha-\beta)\:=\:\sqrt{1+8}} \\ \\ \mapsto\sf{\:(\alpha-\beta)\:=\:\sqrt{9}} \\ \\ \mapsto\sf{\:(\alpha-\beta)\:=\:3.....(3)}$

Add equ(1) and equ(3),

$\mapsto\sf{\:2.\alpha\:=\:4} \\ \\ \mapsto\sf{\:\alpha\:=\:\dfrac{4}{2}} \\ \\ \mapsto\sf{\:\alpha\:=\:2}$

Keep value of α in equ(1),

$\mapsto\sf{\:(2+\beta)\:=\:1} \\ \\ \mapsto\sf{\:\beta\:=\:1-2} \\ \\ \mapsto\sf{\:\beta\:=\:-1}$

$\Large{\underline{\mathfrak{\bf{Thus}}}}$

• Value of 2α = 2×2 = 4
• Value of 2β = 2 × -1 = -2

Now, calculate another equation, whose zeroes are 2α and 2β

Formula of equation:-

$\small\boxed{\sf{\:n^2-(sum\:of\:zeroes)n+(product\:of\:zeroes)\:=\:0}}$

★ Sum of zeroes (2α +2β ) = 4 - 2 = 2

★ product of zeroes = 2α . 2β = 4× -2 = -8

Required equation :-

$\mapsto\sf{\red{\:n^2 - 2n - 8\:=\:0}}$