Enter your answer and show all the steps that you use to solve this problem in the space provided. Find all the zeroes of the polynomial function f(x)=x3−5x2+6x−30. If you use synthetic division, show all three lines of numbers.

Question

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Answer 1

To find the zeroes of the polynomial function f(x) = x^3 - 5x^2 + 6x - 30, we can use synthetic division.

Let's start by finding the possible rational zeroes. They can be determined by taking the factors of the constant term (-30) divided by the factors of the leading coefficient (1).

The factors of -30: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30
The factors of 1: ±1

So, the possible rational zeroes are: ±1, ±2, ±3, ±5, ±6, ±10, ±15, ±30

Now, let's try these values using synthetic division to check if any of them are zeroes:

1 | 1 -5 6 -30
1 -4 2 -28
----------------------
1 -4 2 -28

The remainder is -28, which means x = 1 is not a zero.

-1 | 1 -5 6 -30
-1 6 -12 6
--------------------
1 -6 -6 -24

The remainder is -24, so x = -1 is not a zero either.

2 | 1 -5 6 -30
2 -6 4 -52
--------------------
1 -3 10 -22

The remainder is -22, indicating that x = 2 is not a zero.

-2 | 1 -5 6 -30
-2 14 -20 28
--------------------
1 9 -14 -2

The remainder is -2, so x = -2 is not a zero.

3 | 1 -5 6 -30
3 -6 3 -81
--------------------
1 -2 9 -51

The remainder is -51, which means x = 3 is not a zero either.

-3 | 1 -5 6 -30
-3 24 -78 168
--------------------
1 -8 -72 138

The remainder is 138, indicating that x = -3 is not a zero.

5 | 1 -5 6 -30
5 0 30 180
--------------------
1 0 36 150

The remainder is 150, meaning x = 5 is not a zero.

-5 | 1 -5 6 -30
-5 50 -280 1350
---------------------
1 -10 -274 1320

The remainder is 1320, so x = -5 is also not a zero.

Thus, there are no rational zeroes for the polynomial function f(x) = x^3 - 5x^2 + 6x - 30.