Math, asked by yashdwivedi86521, 11 months ago

Alpha and beta are zeroes of 4x square-5x+1 find the polynomial whose zeroes are alpha square/beta and beta square/alpha

Answers

Answered by SparklingBoy
6

Step-by-step explanation:

Given that α and β are roots of

 {4x}^{2}  - 5x + 1

so now we will expand the above polynomial and calculate alpha and beta.

  \:  \:  \:  \:  \:  \: {4x}^{2}  - 5x + 1  \\  =  {4x}^{2}  - 4x - x + 1 \\    = 4x(x - 1) - 1(x - 1) \\  = x(x - 1)(4x - 1)

Now,

its zeros are 1 and 1/4

so

α=1 and

β=1/4

Now,

now we have to find the polynomial whose zeros are

\frac{ \alpha ^{2}} { \beta}  \:  \:  \:  \:  \: and \:  \:  \:  \frac{ \beta {}^{2} }{ \alpha}

Now

Product of roots of new polynomial = αβ = 1/4

Sum of roots of new polynomial =

 \large  \frac{ { \alpha}^{3} +  { \beta}^{3}  }{ \alpha \beta}   =   \frac{65}{16}

we can write the new polynomial in the form of product and sum of roots as

 {x}^{2}  - sx + p

 =  {x}^{2}  -  \frac{65}{16} x +  \frac{1}{4}  \\  \\ can \: be \: written \: as \:  \\  \\  \frac{1}{16} ( {16x}^{2}  - 65x + 4)

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