Kind a cubic polynomial with the sum, sum of the product of its

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 Multiple Choice QuestionsLong Answer Type

411. Find the zeroes of the quadratic polynomial x2 – 5x + 6, and verify the relationship between the zeroes and the coefficients.
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412. Find the zeroes of x2 –2 and verify the relationship between the zeroes and the coefficients.
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413. Find the zeroes of a quadratic polynomial 5x2 – 4 – 8x and verify the relationship between the zeroes and the coefficients.
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414. Find the zeroes of the following polynomial and verify the relationship between the zeroes and the coefficients
          5 square root of 5 straight y squared plus 20 straight y plus 3 square root of 5
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415.

Verify that 3, -1, fraction numerator negative 1 over denominator 3 end fraction are the zeroes of the cubic polynomial p(x) = 3x3 – 5x2 –11x – 3 then verify the relationship between the zeroes and the coefficients.

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416. Verify that the numbers given along side of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficient in each case.
x3 – 4x2 + 5x – 2, 2, 1, 1.
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417. Kind 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.


Let the quadratic polynomial be ax3 +bx2 +cx + d and its zeroes be α, β and ϒ.
We have,
           straight alpha plus straight beta plus straight gamma space equals space 2 over 1 equals negative fraction numerator Coefficient space of space straight x squared over denominator Coefficient space of space straight x cubed end fraction equals negative straight b over straight a

αβ plus βγ plus γα equals fraction numerator negative 7 over denominator 1 end fraction equals fraction numerator Coefficient space of space straight x over denominator Coefficient space of space straight x cubed end fraction equals straight c over straight a

and   space space space space αβγ equals fraction numerator negative 14 over denominator 1 end fraction equals negative fraction numerator Constant space term over denominator Coefficient space of space straight x cubed end fraction equals fraction numerator negative straight d over denominator straight a end fraction
    When we put a = 1,  then b = -2,  c=-7, d = 14.
Therefore, the required cubic polynomial be x3 – 2x2 – 7x + 14.
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418.

Kind 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 –9, –11, 30 respectively.

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419. Find the zeroes of the quadratic polynomial x2 + 5x + 6 and verify the relationship between the zeroes and the coefficients.
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 Multiple Choice QuestionsShort Answer Type

420. Write a quadratic polynomial, sum of whose zeroes is 2 square root of 3 and their product is 2.
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