two zeros of a cubic polynomial ax3 + 3x2 -bx - 6ax are -1 and -2 .find the third zero and also find the value of a and b?
Let p(x) = ax3 + 3x2 -bx - 6ax
-1 and -2 are the zeroes of the polynomial
p(-1) = ax3 + 3x2 -bx - 6ax
= 5a + b = -3 ..............(1)
p(-2) = 2a + b = -6 .............(2)
From equation (1) and (2), we get
a = 1 and b = -8
Subsitituting the values in the cubic polynomial p(x)
p(x) = x3 + 3x2 + 2x
Now it is given that (x+1)(x+2) is the factor of the polynomial x3 + 3x2 + 2x
⇒ x3 + 3x2 + 2x = (x+1)(x+2)x
So the thord zero is x = 0
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