Question

If α and β are zeroes of the polynomial 2x^2-3x-9. Form a quadratic polynomial whose zeroes are 1/α^2 + 1/ β^2

Answer 1

Answer:

2x2 - 6x +3; -3x -x-4: 1+ 2x - 3x? add the following algebraic expression using both horizontal method and vertical method. did you get the same answer2x2 - 6x +3; -3x -x-4: 1+ 2x - 3x? add the following algebraic expression using both horizontal method and vertical method. did you get the same answer

Answer 2

Answer:

The quadratic polynomial whose roots are   $\frac{1}{\alpha ^2} ,\frac{1}{\beta ^2}$ = 81x²- 45x + 4

Step-by-step explanation:

Given,

α and β are zeroes of the polynomial 2x²-3x-9

To find,

The quadratic polynomial whose zeros are $\frac{1}{\alpha ^2} ,\frac{1}{\beta ^2}$

Recall the concept:

If α and β are zeroes of the polynomial ax²+bx +c, then

α +  β = $\frac{-b}{a}$ and αβ = $\frac{c}{a}$

The equation of the quadratic polynomial if the sum of roots and product of roots are given is

x²- (sum of roots)x +product of roots

Solution:

Since α and β are zeroes of the polynomial 2x²-3x-9,

α +  β =  $\frac{3}{2}$ and αβ = $\frac{-9}{2}$

Required to find the polynomial whose roots are  $\frac{1}{\alpha ^2} ,\frac{1}{\beta ^2}$

Sum of roots =  $\frac{1}{\alpha ^2} +\frac{1}{\beta ^2}= \frac{\alpha ^2 + \beta ^2}{\alpha^2 \beta^2 }$

 $= \frac{(\alpha +\beta )^2 - 2\alpha \beta }{\alpha ^2\beta ^2}$

Substituting the values of α +  β =  $\frac{3}{2}$ and αβ = $\frac{-9}{2}$ we get

Sum of roots $= \frac{(\frac{3}{2} )^2 - 2(\frac{-9}{2}) }{(\frac{-9}{2})^2 }$

$= \frac{\frac{9}{4} + 9 }{\frac{81}{4} }$

= $\frac{45}{81}$

Product of roots = $\frac{1}{\alpha ^2} X\frac{1}{\beta ^2}$

= $(\frac{1}{\alpha} X\frac{1}{\beta})^2$

=$\frac{1}{(\frac{-9}{2})^2 }$

=  $\frac{4}{81}$

Hence the equation whose roots are  $\frac{1}{\alpha ^2} ,\frac{1}{\beta ^2}$ is given by

x²- (sum of roots)x +product of roots

= x²- ( $\frac{45}{81}$)x + $\frac{4}{81}$

Taking LCM = 81

$\frac{1}{81}$[81x²- 45x + 4]

Ignoring  $\frac{1}{81}$, the required polynomial whose roots are   $\frac{1}{\alpha ^2} ,\frac{1}{\beta ^2}$ is given by 81x²- 45x + 4

∴The quadratic polynomial whose roots are   $\frac{1}{\alpha ^2} ,\frac{1}{\beta ^2}$ = 81x²- 45x + 4

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