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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.

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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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Short Answer Type

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

Sum of zeroes = $2 square root of 3$
and product of zeroes = 2
We know that a quadratic polynomial when the sum and product of its zeroes are given by
f(x) = k {x² – (sum of zeroes) x + product of zeroes}
where k is a constant
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