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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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424.

If α and β are the zeroes of the quadratic polynomial ax² + bx + c. Find the value of α²– β².

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

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

Now,

straight alpha squared over straight beta plus straight beta squared over straight alpha equals fraction numerator straight alpha cubed plus straight beta cubed over denominator βα end fraction

space equals fraction numerator left parenthesis straight alpha plus straight beta right parenthesis cubed minus 3 αβ left parenthesis straight alpha plus straight beta right parenthesis over denominator αβ end fraction

space space space equals fraction numerator open parentheses begin display style fraction numerator negative straight b over denominator straight a end fraction end style close parentheses cubed minus 3 open parentheses begin display style straight c over straight a end style close parentheses open parentheses begin display style fraction numerator negative straight b over denominator straight a end fraction end style close parentheses over denominator begin display style straight c over straight a end style end fraction

equals fraction numerator begin display style fraction numerator negative straight b cubed over denominator straight a cubed end fraction end style plus begin display style fraction numerator 3 bc over denominator straight a squared end fraction end style over denominator begin display style straight c over straight a end style end fraction equals fraction numerator begin display style fraction numerator negative straight b cubed plus 3 abc over denominator straight a cubed end fraction end style over denominator begin display style straight c over straight a end style end fraction

space space space space space space equals space fraction numerator 3 abc minus straight b cubed over denominator straight a cubed end fraction cross times straight a over straight c equals fraction numerator 3 abc minus straight b cubed over denominator straight a squared straight c end fraction.

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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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