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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.

Since, most general form of cubic polynomial is ax³ + bx² + cx + d When, we compare the given cubic polynomial with the general form, we get
a = 3, b = –5
c = – 11 and d = – 3
Let P(x) ax^3, + bx² + cx + d
Now, P(3) = 3 (3)³ + (– 5) (3)²
+ (-11) (3) + (-3)
= 81 - 45 - 33 - 3
= 0

space space straight P left parenthesis negative 1 right parenthesis space equals space 3 left parenthesis negative 1 right parenthesis cubed plus left parenthesis negative 5 right parenthesis left parenthesis negative 1 right parenthesis squared space space space space space space space space space space space space space space space plus left parenthesis negative 11 right parenthesis space left parenthesis negative 1 right parenthesis space plus space left parenthesis negative 3 right parenthesis

space equals 3 left parenthesis negative 1 right parenthesis minus 5 plus 11 minus 3 space equals negative 3 minus 5 plus 11 minus 3 space equals negative 11 plus 11 equals 0

and

straight P open parentheses negative 1 third close parentheses space equals space 3 open parentheses negative 1 third close parentheses cubed plus left parenthesis 5 right parenthesis open parentheses negative 1 third close parentheses squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space

plus left parenthesis negative 11 right parenthesis open parentheses negative 1 third close parentheses plus left parenthesis negative 3 right parenthesis

equals 3 cross times fraction numerator negative 1 over denominator 27 end fraction minus 5 cross times 1 over 9 plus 11 over 3 minus 3 over 1

equals negative 1 over 9 minus 5 over 9 plus 11 over 3 minus 3 over 1

equals fraction numerator negative 1 minus 5 plus 33 minus 27 over denominator 9 end fraction

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Therefore, the given values 3, -1 and

fraction numerator negative 1 over denominator 3 end fraction

are the zeroes of 3x³ – 5x² – 11x – 3.
Verification of relationship between zeroes and the coefficients the zeroes of 3x³ – 5x² – 11x – 3.
Since, most general form of cubic polynomial is ax3 + bx2 + cx + d

Since, most general form of cubic polynomial is ax3 + bx2 + cx + d

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

equals negative fraction numerator Constant space term over denominator Coefficient space of space straight x cubed end fraction.

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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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