Question

if alpha and beta are the zeros of the polynomial x²+4x+2 then find a polynomial whose zeroes are 2 alpha + beta and alpha + 2beta​

Answer 1

$\large\underline\blue{\bold{Given \:  :-  }}$

$\rm : \implies \: \alpha \: and \: \beta \: are \: zeroes \: of \: {x}^{2} + 4x + 2$

$\large\underline\blue{\bold{To \: Find :-  }}$

$\rm : \implies \:a \: polynomial \: having \: zeroes \: 2 \alpha + \beta \: and \: 2 \beta + \alpha$

$\large\underline\purple{\bold{Solution :- }}$

Now,

We know that,

$\boxed{\purple{\tt Sum\ of\ the\ zeroes=\frac{-b}{a}}}$

OR

$\boxed{\red{\sf Sum\ of\ the\ zeroes=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}}$

And

$\boxed{\purple{\tt Product\ of\ the\ zeroes=\frac{c}{a}}}$

OR

$\boxed{\red{\sf Product\ of\ the\ zeroes=\frac{Constant}{coefficient\ of\ x^{2}}}}$

So,

$\rightarrow \rm \: \alpha + \beta = \frac{-b}{a}$

$\bigstar \: \: \boxed{ \pink{ \rm : \implies \: \alpha + \beta \: = - \: 4}}$

and

$\rightarrow \rm \: \alpha \beta = \frac{c}{a}$

$\bigstar \: \: \boxed{ \pink{ \rm : \implies \: \alpha \beta \: = \: 2}}$

Now,

To Find a quadratic Polynomial having zeroes

$\rm \: 2 \alpha + \beta \: \: and \: \: 2\beta + \alpha$

We know,

If S represents the Sum of zeroes and P represents the Product of zeroes, then required quadratic polynomial is given by

$\bigstar \: \: \boxed{ \pink{ \rm : \implies \:f(x )\: = k \:( {x}^{2} - Sx + P) \: \: where \: k \: \ne \: 0}}$

So,

$\rightarrow \rm \: Sum \: of \: zeroes \:(S) = 2 \alpha + \beta + 2\beta + \alpha$

$\rm : \implies \:S \: = 3 \alpha + 3 \beta$

$\rm : \implies \:S \: = \: 3( \alpha + \beta )$

$\rm : \implies \:S \: = \: 3 \times ( - 4)$

$\bigstar \: \: \boxed{ \pink{ \rm : \implies \:S \: = - \: 12}}$

Also,

$\rightarrow \rm \: Product \: of \: zeroes \: (P) = (2 \alpha + \beta )(2 \beta + \alpha )$

$\rm : \implies \:P = {2 \alpha }^{2} + 4\alpha \beta + {2 \beta }^{2} + \alpha \beta$

$\rm : \implies \:P \: = 2( { \alpha }^{2} + 2 \alpha \beta + { \beta }^{2} ) + \alpha \beta$

$\rm : \implies \:P \: = 2 {( \alpha + \beta) }^{2} + 2$

$\rm : \implies \:P \: = 2 {( - 4)}^{2} + 2$

$\bigstar \: \: \boxed{ \pink{ \rm : \implies \:P\: = \: 34}}$

Hence,

The required Quadratic polynomial is given by

$\rightarrow \rm \: f(x) \: = \: k \: ( {x}^{2} - Sx + P) \: \: where \: k \: \ne \: 0$

$\rm : \implies \:f(x) \: = \: k( {x}^{2} - ( - 12)x + 34) \: \: where \: k \: \ne \: 0$

$\bigstar \: \: \boxed{ \pink{ \rm : \implies \:f(x) \: = \: k( {x}^{2} + 12x + 34) \: \: where \: k \: \ne \: 0}}$