Verify that 3, -1 and -⅓ are the zeroes of the cubic polynomial p(x) = 3x3 - 5x2 -11x - 3 and also verify the relation between the zeroes and coefficients.
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Step-by-step explanation:
p(3) = 3(3)^3 - 5(3)^2 - 11(3) -3
= 3(27) - 5(9) - 33 -3
= 81 - 45 - 33 -3
= 81 - 81
= 0
p(-1) = 3(-1)^3 - 5(-1)^2 -11(-1) -3
= 3(-1) -5(1) + 11 - 3
= - 3 - 5 + 11 - 3
= - 11 + 11
=0
p(-1/3) = 3( -1/3)^3 - 5(-1/3)^2 - 11(-1/3) - 3
= 3(-1/27) - 5(1/9) + 11/3 - 3
= -1/9 - 5/9 + 11/3 -3
= -6/9 + 11/3 -3
= -6 + 33 - 27 / 9
= 0
as they all when substituted give result zero 3, -1 , -1/3 are roots of p(x)
sum of roots = 3 - 1 - 1/3 = 5/3 = -b/a
sum of roots two at a time= -3 + 1/3 - 1 = - 11/3 =c/a
product of roots = 1 = -d/a
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