Find the zeros of the polynomial f(x)=x3+5x2-2x-24 if it is given that product of its two zeros is 12
Answers
Step-by-step explanation:
Given: alpha , beta and gamma are the zeros of the polynomial f(x)=x3+5x2-2x-24
Condition: alpha × beta= 12 -(1)
To find : gamma
We know that,
The product of the zeros =-Constant term upon coefficient of x3
alpha×beta×gamma=-(-24)upon 1
12×gamma=24
from equation (1)
gamma=24divided by 12
implies gamma = 2 -(2)
If x= 2
then x -2
g(x) = x-2
divide f(x) by g(x)
x-2 divided by x3+5x2-2x-24
your answer will be x2 +7x+12
By using splitting the middle term method
x2+7x+12 12×x2=12x2 +4x×3x
x2+4x+3x+12
x(x+4)3(x+3)
(x+3)(x+4)
x+3=0 x+4=0
x=-3 x=-4
Therefore-3 ,-4 and 2 are the zeros of the polynomial.
alpha=-3
beta=-4
gamma=2
Verification:
-3×-4=12
12=12
Therefore L.H.S.=R.H.S.