If one zero of the cubic polynomial f(x)= 3x^3 - 5x^2- 11x -3 is -1/3 then find the other zeroes
Answers
Given that
f(x) = 3x³ - 5x² - 11x - 3
One of its zero =
To find
All the remaining zeroes = ?
Assumption
Let's assume that
The three zeroes of f(x) are :
As we know
There exists a relationship between the co-efficients and the zeroes of a polynomial.
As f(x) is a cubic polynomial (as the degree is 3) there will be maximum 3 zeroes and not more than that.
In some rare cases, a cubic polynomial can also have only 1 zero or it can also have 2 zeroes.
The following are the relationship between the co-efficients and the zeroes of a cubic polynomial :
(i)
(ii)
(iii)
Now, we'll find the zeroes of f(x).
If one zero is
3x + 1
Divide f(x) by 3x + 1 to obtain other zeroes.
q(x) = x² + 2x - 3
r(x) = 0
Now, we've to factorise q(x) to obtain our zeroes.
x² + 2x - 3 = 0
The factors of - 3 which gives us the sum as 2 are
x² + 2x - 3 = 0
x² + 3x - x - 3 = 0
x(x + 3) - 1(x + 3) = 0
(x + 3) (x - 1) = 0
(x + 3) = 0
x + 3 = 0
x - 1 = 0
Therefore, the zeroes of the f(x) are 1, - 3 and