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411. Find the zeroes of the quadratic polynomial x² – 5x + 6, and verify the relationship between the zeroes and the coefficients.

We have x² – 5x + 6 – x² – 3x – 2x + 6
= (x – 3) (x – 2)
So, the value of x² – 5x + 6 is zero when x - 3 = 0 or x – 2 = 0
i.e., when x = 3 or x = 2
Therefore, the zeroes of x² – 5x + 6 are 3 and 2.
Sum of zeroes =

3 plus 2 space equals space 5 over 1

fraction numerator negative left parenthesis Coefficient space of space straight x right parenthesis over denominator Coefficient space of space straight x squared end fraction

Product of zeroes

equals 3 cross times 2 equals 6 over 1 equals fraction numerator Constant space term over denominator Coefficient space of space straight x squared end fraction

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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.

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