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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.

The general form of cubic polynomial is ax³ + bx² + cx + d when, we compare the given cubic polynomial x³ - 4x² + 5x – 2 with general form, we get
a = 1, b = -4
c = 5 and d = -2

Let space space space space space space space space space space space space space space space space space space space space space space straight P left parenthesis straight x right parenthesis space equals space ax cubed plus bx squared plus cx plus straight d 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 straight P left parenthesis 2 right parenthesis space equals space 1 left parenthesis 2 right parenthesis cubed plus left parenthesis negative 4 right parenthesis space left parenthesis 2 right parenthesis squared plus space 5 space left parenthesis 2 right parenthesis space plus space left parenthesis negative 2 right parenthesis 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 space equals space 8 minus 16 plus 10 minus 2 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 space equals negative 8 plus 8 equals 0

straight P left parenthesis 1 right parenthesis space equals space 1 left parenthesis 1 right parenthesis cubed plus left parenthesis negative 4 right parenthesis left parenthesis 1 right parenthesis 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 space left parenthesis 5 right parenthesis space left parenthesis 1 right parenthesis space plus space left parenthesis negative 2 right parenthesis space space space space space space space space space equals space 1 minus 4 plus 5 minus 2 space space space space space space space space space equals space 6 minus 6 space equals space 0

and

straight P left parenthesis 1 right parenthesis space equals space 1 left parenthesis 1 right parenthesis cubed plus left parenthesis negative 4 right parenthesis left parenthesis 1 right parenthesis squared plus left parenthesis 5 right parenthesis space left parenthesis 1 right parenthesis space plus space left parenthesis negative 2 right parenthesis

space equals 1 minus 4 plus 5 minus 2 space space equals space 6 minus 6 space equals space 0

Verification of relationship between zeroes and coefficients.
Let

space space space space straight alpha space equals space 2 comma space space space space straight beta equals 1 comma space space straight gamma equals 1

Now,

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equals negative fraction numerator Coefficient space of space straight x squared over denominator Coefficient space of space straight x cubed end fraction

αβ plus βγ plus γα space equals space left parenthesis 2 right parenthesis left parenthesis 1 right parenthesis plus left parenthesis 1 right parenthesis left parenthesis 1 right parenthesis plus left parenthesis 1 right parenthesis left parenthesis 2 right parenthesis

equals 2 plus 1 plus 2 equals 5 over 1

equals fraction numerator Coefficient space of space straight x over denominator Coefficient space of space straight x cubed end fraction

αβγ space equals space left parenthesis 2 right parenthesis space left parenthesis 1 right parenthesis space left parenthesis 1 right parenthesis space equals space 2

equals fraction numerator negative left parenthesis negative 2 right parenthesis over denominator 1 end fraction

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

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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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