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351. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also, verify the relationship between the zeroes and coefficients in each case:
x³ - 4x² + 5x - 2; 2, 1, 1

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Short Answer Type

352. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, -7, -14 respectively.

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Long Answer Type

353. If the zeroes of the polynomial x³ – 3x²+ x + 1 are (a – b), a, (a + b), find a and b.

Since a – b, a and a + b are the zeroes of f(x).

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space space space space space straight a minus straight b plus straight a plus straight a plus straight b equals negative fraction numerator Coeff. of space straight x squared over denominator Coeff. space of space straight x cubed end fraction

equals fraction numerator negative left parenthesis negative 3 right parenthesis over denominator 1 end fraction

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space space space 3 straight a equals space 3 space rightwards double arrow space space space straight a space equals space 1

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(a - b) a(a + b) =

space space space minus fraction numerator Constsnt space term over denominator Coeff. space of space straight x cubed end fraction

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straight b squared space equals space 2 space rightwards double arrow space space straight b equals plus-or-minus square root of 2

Hence,

straight a equals 1 comma space space straight b space equals space plus-or-minus square root of 2.

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354. If two zeroes of the polynomial x⁴ – 26x³-26x² + 138x –35 are 2 ±

square root of 3

, find other two zeroes.

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355. If the polynomial x⁴ – 6x³ + 16x² – 25x + 10 is divided by another polynomial x² - 2x + k, the remainder comes out to be x + a, find ‘k’ and ‘a’.

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