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421. Find a quadratic polynomial whose zeroes are 1 and (-3). Verify the relation between the coefficients and zeroes of the polynomial.

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422. Find the zeroes of the quadratic polynomial 4x² – 4x – 3 and verify the relation between the zeroes and its coefficients.

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423. If α and β are the zeroes of the polynomial ax² + bx + c then find
(a)

straight alpha over straight beta plus straight beta over straight alpha

(b)

space space straight alpha squared plus straight beta squared

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

space space space space space straight alpha plus straight beta equals fraction numerator negative straight b over denominator straight a end fraction space and space straight alpha. space straight beta space equals space straight c over straight a

Now,

straight alpha squared minus straight beta squared space equals space left parenthesis straight alpha plus straight beta right parenthesis space left parenthesis straight alpha minus straight beta right parenthesis

space space space space equals left parenthesis straight alpha plus straight beta right parenthesis space square root of left parenthesis straight alpha plus straight beta right parenthesis squared minus 4 αβ end root

equals open parentheses fraction numerator negative straight b over denominator straight a end fraction close parentheses square root of open parentheses fraction numerator negative straight b over denominator straight a end fraction close parentheses squared minus 4 cross times straight c over straight a end root

equals space space open parentheses fraction numerator negative straight b over denominator straight a end fraction close parentheses square root of straight b squared over straight a squared minus fraction numerator 4 straight c over denominator straight a end fraction end root

space space space equals space fraction numerator negative straight b over denominator straight a end fraction square root of fraction numerator straight b squared minus 4 ac over denominator straight a squared end fraction end root equals fraction numerator negative straight b square root of straight b squared minus 4 ac end root over denominator straight a squared end fraction.

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425. If α and β are the zeroes of the quadratic polynomial ax² + bx + c, then find the value of α³ + β³.

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426. If α and β are the zeroes of quadratic polynomial ax² + bx + c, then find

straight alpha squared over straight beta plus straight beta squared over straight alpha.

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427. If α and β are the zeroes of the quadratic polynomial P(x) = Kx² + 4x + 4 such that α² + β² = 24, find the value of K.

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428. If α and β are the zeroes of quadratic polynomial x² + x – 2. Find the value of (α^–1 + β^–1).

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429. Find a quadratic polynomial when the sum and product of its zeroes respectively
(i)

space 1 fourth comma space minus 1

(ii)

square root of 2 comma space fraction numerator negative 1 over denominator 3 end fraction

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430. If α, β are the zeroes of the quadratic polynomial 2x²– 3x – 5, form a polynomial whose zeroes are 2α + 1 and 2β + 1.

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