Question

the sum of an product of polynomial respectively root 2,1/3 find the equation
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Answer 1

Answer:

let,a and b are roots

a+b =2

ab = 1/3

eqn =x^2-x(2)+1/3

x^2-2x+1/3

3x^2-6x+1

Answer 2

Question :

The sum of an product of polynomial respectively √2,1/3 find the equation.

Theory :

if $\sf\alpha \: and \beta$ are the zeros of a quadratic polynomial

f(x) . Then the polynomial f x is given by

$\sf \:f(x) = k(x {}^{2} - ( \alpha + \beta )x + \alpha \beta )$

or

$\sf \: f(x) = k(x {}^{2} - (sum \: of \: the \: zeroes)x + product \: of \: the \: zeroes)$

Solution :

Let $\sf \alpha \: and \: \beta$ are the zeros of the required polynomial.It is given that

$\sf \alpha + \beta = \sqrt{2}$

$\sf \: and \: \alpha \beta = \dfrac{1}{3}$

The quadratic polynomial is

$\sf \: f(x) = x {}^{2} - ( \alpha + \beta )x + \alpha \beta$

$\sf \implies \: f(x) = x {}^{2} - \sqrt{2}x + \dfrac{1}{3}$

Taking the LCM:-

$\sf\implies\:f(x)=\dfrac{3x^{2}-3\sqrt{2}x+1}{3}$