Question

if alpha and beta are the zeros of polynomial x^2 + 4x +3 form the polynomial whose zeros are 1 + alpha/beta and 1 + beta / alpha

Answer 1

◢GIVEN EQUATION

=> x² + 4x + 3

◢AND

$\alpha \: \: AND \: \: \beta$

● ARE ZEROS OF THE POLYNOMIAL.

_______________________________

◢NOW,

● FACTORISING

$= > {x}^{2} + 4x + 3 = 0 \\ \\ = > {x}^{2} + 3x + x + 3 = 0 \\ \\ = > x(x + 3) + 1(x + 3) = 0 \\ \\ = > (x + 1)(x + 3) = 0$
_______________________________

◢THEN,

$= > (x + 1) = 0 \\ \\ = > x = - 1$

◢AND

$= > (x + 3) = 0 \\ \\ = > x = - 3$

_______________________________

◢SO,

$= > \alpha = - 3$

◢AND

$= > \beta = - 1$

_______________________________

● THE ZEROS OF THE NEW EQUATION ARE

$\frac{1 + \alpha }{ \beta } \: \: \: \: \: AND \: \: \: \: \: \frac{1 + \beta }{ \gamma }$

◢THEN,

$= > \frac{1 + \alpha }{ \beta } = \frac{1 + ( - 3)}{ - 1} = \frac{ - 2}{ - 1} = 2$

◢AND

$= > \frac{1 + \beta }{ \alpha } = \frac{1 + ( - 1)}{ - 3} = \frac{0}{3} = 0$
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● SO THE ZEROS OF THE EQUATION ARE "2" AND "0"
_______________________________

● SUM OF THE ZEROS

$= > 0 + 2 = 2$

◢AND

● PRODUCT OF THE ZEROS

$= > 0 \times 2 = 0$
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◢THEN,

● TO FROM THE QUADRATIC EQUATION WE HAVE FORMULA AS :-

$= > {x}^{2} - (SUM \: \: OF \: \: ZEROS)x \: \\ + \: (PRODUCT \: \: OF \: \: ZEROS) = 0$

PLUG THE VALUES

$= > {x}^{2} - 2x + 0 = 0 \\ \\ = > {x}^{2} - 2x = 0$
_____________________[ANSWER]

SO THE REQUARIED QUADRATIC EQUATION IS :- => x²-2x = 0

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☆$BE \: \: BRAINLY$☆

Answer 2

$\color{red}{ \bf{hey}}$

$\color{blue}{ \underline{here \: is \: the \: answer}}$
⬇⤵⬇⏬⏬⏬

given: p(x)= x²+4x+3

zeroes are

1+alpha/beta

and 1+beta/alpha

first factorize p(x)

⏩ x²+4x+3

⏩ x²+3x+x+3

⏩ x(x+3)+1(x+3)

⏩ (x+1)(x+3)

x+1=0

x=-1

x+3=0

x=-3

hence ,
$\alpha = - 3$
and
$\beta = - 1$

1+alpha/beta= 1+(-3)/-1

1-3/-1

-2/-1

=> 2

and

1+beta/alpha= 1+(-1)/-3

0..

hence the polynomial is

x²-(sum of zeroes )x+product of zeroes ..

⏩x²-(0+2)x+0

⏩x²-2x+0

hence , the polynomial is

$\color{green}{ \bf{x {}^{2} - 2x}}$
$\color{orange}{ \underline{hope \: it \: helps}}$
$\color{voilet}{ \boxed{thanks}}$