if alpha and beta are zeroes of polynomial p(x)=2x2-7x+5. Find a polynomial whose zeroes are 2alpha+1 and 2beta+3

Answer :

Given :α and  β   are two zeros of the polynomial

Equation  f ( x ) = 2 x2- 7 x  + 5
So,
We know from relationship between zeros and coefficient ,

Sum of zeros  = -coefficient xcoefficient x2 , So

α + β72            --- ( 1 )
And

Product of zeros  = constant termcoefficient x2 , So
αβ  =  52               ------ ( 2 )

Taking Whole square of equation 1 , we get 

( α + β  )2494 

And

( αβ  )2  + 4αβ494   , Substitute value from equation 2 , we get

( αβ  )2  + 4 ( 52 ) =  494 

( αβ  )2  + 10 =  494 

( αβ  )2  = 494 - 10

( αβ  )2  = 49 - 404

( αβ  )2  = 94

αβ   =  32                        ---- ( 3 )

Add equation 1 and 3 , We get

2α   =   5

α   =  52  , Substitute that value in equation 3 we get

52  - β = 32    , So

β  = 1

So,

We  get  α   = 52   and β  = 1

Then

2α  + 1 = 2 ( 52  ) + 1   =5 + 1  = 6
And
β  + 3 = 2 ( 1  ) + 3   = 2 + 3  = 5

Then

Sum of zeros of required polynomial  = (2α  + 1 )+  (2  β  + 3 ) =  6 +  5 = 11
And

Product of zeros of required polynomial  = (2α  + 1 )(2  β  + 3 )= ( 6 ) ( 5 ) = 30

And we know formula for polynomial when some of zeros and product of zeros we know :


Polynomial  =  k [ x2  - ( Sum of zeros ) x  + ( Product of zeros ) ]   , Here k is any non zero real number.

Substitute values , we get

Polynomial  =  k [ x2  - ( 11  ) x  + ( 30 ) ] 

Polynomial  =  k [ x2  - 11 x  + 30] 

Then

Quadratic polynomial  =    k [ x2  - 11 x  + 30]    =   x2  - 11 x  + 30    [on taking k = 1]                              ( Ans )

  • -63
sum = alpha + beta = 7/2
product = alpha * beta = 5/2
now 
sum of other zeroes = 2 alpha + 1 + 2beta + 3lpha + beta)
                               = 2( alpha + beta ) + 4
                             = 2 * 7/2 + 4
                            = 11
product = (2 alpha + 1) ( 2 beta +3)
                =2(alpha + beta ) + 6alpha + 2beta + 3
                =2(alpha + beta ) + 2(3alpha + beta ) + 3
               = 2*7/2 + 6 * 7/2 + 3
             = 20
then g (x) = x2 - 11x + 20
  • 28
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