Question

if alpha and beta are the zeros of quadratic polynomial f x is equal to 2 x square minus 5 x + 7 find a polynomial whose zeros are 2 alpha + 3 beta and 3 alpha plus 2 beta

Answer 1

Hey there !!!!!!!!

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f(x) = 2x²-5x+7

2x²-5x+7=0

x²-5x/2+7/2= 0 

A quadratic equation whose roots are α,β can be written as x²-(α+β)x+αβ

Comparing x²-5x/2+7/2= 0  with x²-(α+β)x+αβ 

α+β = 5/2   αβ= 7/2

Now ,

According to question a quadratic equation has 2α+3β & 3α+2β as the roots of equation 

So equation is 

P(x) = x²-(2α+3β+3α+2β)x+(2α+3β)(3α+2β)

       = x²-(5α+5β)x+(2α+2β+β)(α+2α+2β)

      =  x²-5(α+β)x+(2(α+β)+β)(α+2(α+β))

But   α+β = 5/2   αβ= 7/2

So,

=  x²-5(α+β)x+(2(α+β)+β)(α+2(α+β))

=x²-5(5x)/2 +( 5+β)(α+5)

= x²-25x/2 +(25+5(α+β)+αβ)

=x²-25x/2+(25+5*5/2+7/2)

=2x²-25x+82

So required equation is 2x²-25x+82

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Hope this helped you...............