Using complete sentences, explain how to find the zeros of the function f(x) = 2x^3 – 9x + 3
To find the zeros of the function f(x) = 2x^3 - 9x + 3, we need to equate f(x) to 0 and solve the equation we get.
Here we have to solve 2x^3 - 9x + 3 = 0
As the highest power of x among all the the terms is 3, we will get 3 values for x which satisfy the equation.
The function has 3 zeros.
To find the roots of the cubic equation that we have got, the following formula can be used
2x^3 - 9x + 3 = 0
=> x^3 - (9/2)x + 3/2 = 0
a = -9/2 and b = 3/2
Now substitute the values of a and b in the following to determine the three zeros:
Let A = cuberoot(-b/2+sqrt(b^2/4+a^3/27))
and B = cuberoot(-b/2-sqrt(b^2/4+a^3/27))
Then the three solutions are:
x1 = A + B
x2 = -(A+B)/2 + (A-B)sqrt(-3)/2
and x3= -(A+B)/2 - (A-B)sqrt(-3)/2