Question

Two zeroes of cubic polynomial ax^3 +3x^2-bx-6 are -1 and -2. Find the 3rd zero and the value of a and b.
Plz ans.......

Answer 1

$\underline{\large{\red{\bf{QuesTion}}}}$

Two zeroes of cubic polynomial ax³+3x²-bx-6 are -1 and -2 . Find the 3rd zero and the value of a and b.

$\underline{\large{\blue{\bf{AnswEr}}}}$

on putting (-1) in place of x in given cubic polynomial the polynomial should equate with zero,

ax³+3x²-bx-6 = 0 when( x = -1)

a(-1)³+3(-1)²-b(-1)-6=0

-a+3+b-6=0

b-a-3=0

b=a+3 .......eqn(1)

also, on putting x = -2 it should equate with zero

so,

a(-2)³+3(-2)²-b(-2)-6=0

-8a+12+2b-6=0

2b-8a+6=0

(using eqn(1))

2(a+3)-8a+6 = 0

2a+6-8a+6=0

-6a = -12

$\boxed{\tt a=2}$

putting value of a in eqn(1)

b = a+3

b = 2 +3

$\boxed{\tt b = 5}$

so, the cubic polynomial will be

2x³+3x²-5x-6

_____________________________________

let, the third zero of cubic polynomial be 'm'

Now ,

in the cubic polynomial

$\tt sum\:of\:three\:zeroes=\frac{-(coefficient\:of\:x^2)}{coefficient\:of\:x^3}$

so,

-1 + (-2) + m = -3 / 2

-3 + m = -3 / 2

$\tt m = \frac{-3}{2}+3\\\\\tt m = \frac{-3+6}{2}\\\\\tt m = \frac{3}{2}\\\\\boxed{\tt third\:zero\:of\:polynomial\:is\:3/2}$