Find all the zeroes of the polynomial function f(x)=x^3−5x^2+6x−30
To find the zeroes of a polynomial, we set the function equal to zero and solve for x.
So, we have:
x^3−5x^2+6x−30=0
There are several methods to solve this equation, but one possible way is by factoring. To factor this polynomial, we can start by trying different values for x such as 1, -1, 2, -2 and so on, to check if any of these values are zeroes of the polynomial.
Let's check the value x = 1:
(1)^3−5(1)^2+6(1)−30=1−5+6−30=-28
The value is not zero, so 1 is not a zero of the polynomial. Let's try another value.
Let's check the value x = -1:
(-1)^3−5(-1)^2+6(-1)−30=-1−5-6−30=-42
The value is not zero, so -1 is not a zero of the polynomial. Let's try another value.
Let's check the value x = 2:
(2)^3−5(2)^2+6(2)−30=8−20+12−30=-30
The value is zero, so 2 is a zero of the polynomial. This means (x-2) is a factor of the polynomial. We can use long division or synthetic division to divide the polynomial by (x-2) to find the remaining zeros.
Using long division, we have:
x^2 - 3x + 15
___________________
x - 2 | x^3 - 5x^2 + 6x - 30
-(x^3 - 2x^2)
-----------------
-3x^2 + 6x
3x^2 - 6x
-------------
0
So, we find that x^2 - 3x + 15 is the remaining factor after dividing by (x-2). To solve for the remaining zeros, we can set this factor equal to zero:
x^2 - 3x + 15 = 0
This quadratic equation has no real solutions since its discriminant (b^2 - 4ac) is negative. Therefore, there are no more real zeros of the polynomial function f(x)=x^3−5x^2+6x−30.