Question

If α and β are the zeros of the quadratic polynomial f(x) = 2x² -5x + 7, find a polynomial whose zeros

are 2α+ 3β and 3α+ 2β?​

Answer 1

$\large\underline{\bf{Solution-}}$

Given that

• α and β are the zeros of the quadratic polynomial f(x) = 2x² -5x + 7

We know,

$\boxed{\red{\sf Sum\ of\ the\ zeroes=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{2}}}}$

$\rm :\implies\: \alpha + \beta = - \dfrac{( - 5)}{2} = \dfrac{5}{2} - - (1)$

And

$\boxed{\red{\sf Product\ of\ the\ zeroes=\frac{Constant}{coefficient\ of\ x^{2}}}}$

$\rm :\implies\: \alpha \beta = \dfrac{7}{2} - - (2)$

Now,

To find a quadratic polynomial having zeroes 2α+ 3β and 3α+ 2β.

Consider,

$\rm :\longmapsto\:Sum \: of \: zeroes \:$

$\rm :\longmapsto\:2 \alpha + 3 \beta + 3 \alpha + 2 \beta$

$\rm \: \: = \: \: 5\alpha + 5\beta$

$\rm \: \: = \: \: 5(\alpha + \beta)$

$\rm \: \: = \: \: 5 \times \dfrac{5}{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \: using \: (1)\bigg \}}$

$\rm \: \: = \: \: \dfrac{25}{2}$

Consider,

$\rm :\longmapsto\:Product \: of \: zeroes$

$\rm \: \: = \: \: (2 \alpha + 3 \beta )(2 \beta + 3 \alpha )$

$\rm \: \: = \: \: 4 \alpha \beta + {6 \alpha }^{2} + {6 \beta }^{2} + 9 \alpha \beta$

$\rm \: \: = \: \: {6 \alpha }^{2} + {6 \beta }^{2} + 12 \alpha \beta + \alpha \beta$

$\rm \: \: = \: \: 6({ \alpha }^{2} + { \beta }^{2} + 2 \alpha \beta) + \alpha \beta$

$\rm \: \: = \: \: 6({\alpha + \beta ) }^{2} + \alpha \beta$

$\rm \: \: = \: \: 6 \times \dfrac{25}{4} + \dfrac{7}{2}$

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \red{\bigg \{ \because \: using \: (1) \: and \: (2)\bigg \}}$

$\rm \: \: = \: \: \dfrac{75}{2} + \dfrac{7}{2}$

$\rm \: \: = \: \: \dfrac{82}{2}$

$\rm \: \: = \: \: 41$

So,

The required Quadratic polynomial whose zeroes are 2α+ 3β and 3α+ 2β is given by

$\bf :\longmapsto\:p(x) = k( {x}^{2} - Sx + P) \: where \: k \ne \: 0$

where,

• S is sum of zeroes

• P is product of zeroes

Hence,

$\bf :\longmapsto\:p(x) = k\bigg( {x}^{2} - \dfrac{25}{2}x + 41\bigg) \: where \: k \ne \: 0$