Question

7. α and β are zeroes of the quadratic polynomial x2 – 6x + y. Find the value of ‘y’ if 3α + 2β = 20.

8. If the zeroes of the polynomial x3 – 3x2 + x + 1 are a – b, a, a + b, then find the value of a and b.

9. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes, respectively.

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Answer 1

Answer :- 7

Given that

$\sf \: \alpha \: and \: \beta \: are \: the \: zeroes \: of \: {x}^{2} - 6x + y$

We know that,

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

$\bf\implies \: \alpha + \beta = - \dfrac{( - 6)}{1} = 6 - - - (1)$

Also,

Given that,

$\rm :\longmapsto\:3 \alpha + 2 \beta = 20$

$\rm :\longmapsto\: \alpha + 2 \alpha + 2 \beta = 20$

$\rm :\longmapsto\: \alpha + 2( \alpha + \beta) = 20$

$\rm :\longmapsto\: \alpha + 2 \times 6 = 20$

$\rm :\longmapsto\: \alpha + 12 = 20$

$\rm :\longmapsto\: \alpha = 20 - 1$

$\bf\implies \: \alpha = 8$

$\sf \: Since, \: \alpha \:is \: the \: zeroes \: of \: {x}^{2} - 6x + y$

$\sf \: \therefore \: \alpha \: = 8 \: must \: satisfy \: {x}^{2} - 6x + y$

$\rm :\implies\: {8}^{2} - 6 \times 8 + y = 0$

$\rm :\implies\: 64 - 48 + y = 0$

$\rm :\implies\: 16+ y = 0$

$\rm :\implies\: y = - \: 16$

Answer :- 8

Given that

$\sf \: a - b,a,a + b \: are \: zeroes \: of \: {x}^{3} - {3x}^{2} + x + 1$

We know that,

$\boxed{\red{\sf Sum\ of\ the\ zeroes=\frac{-coefficient\ of\ x}{coefficient\ of\ x^{3}}}}$

$\rm :\implies\:a - b + a + a + b = - \dfrac{( - 3)}{1}$

$\rm :\implies\:3a= 3$

$\pink{\bf\implies \:a = 1}$

Also,

We know that,

$\boxed{\red{\sf Product\ of\ the\ zeroes= - \: \frac{Constant}{coefficient\ of\ x^{3}}}}$

$\rm :\implies\:(a - b)(a)(a + b) = - \dfrac{(1)}{1}$

Put a = 1, we get

$\rm :\longmapsto\:(1 - b)(1)(1 + b) = - 1$

$\rm :\longmapsto\:1 - {b}^{2} = - 1$

$\rm :\longmapsto\:{b}^{2} = 2$

$\purple{\bf\implies \:b = \pm \: \sqrt{2} }$

Answer :- 9

Let

• 'S' represents the Sum of the zeroes of a quadratic polynomial.

and

Let

• 'P' represents the Product of the zeroes of a quadratic polynomial.

Then,

Required Quadratic polynomial is

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