Factorising x^3 - 3x^2 + 9x - 5

To factorise the given expression, we will use the factor theorem and synthetic division method.

Step 1: Identify possible factors
The first step in factorising a polynomial is to identify any possible factors. In this case, the leading coefficient is 1, and the constant term is -5. Therefore, the possible factors can be obtained by finding the factors of -5.

The factors of -5 are ±1 and ±5.

Step 2: Use synthetic division to test the factors
We will use synthetic division to test the factors and find the one that results in a remainder of zero.

We start by testing x = 1.

1 | 1 -3 9 -5
| 1 -2 7
-------------
1 -2 7 2

The remainder is 2, which means x = 1 is not a factor.

Next, we test x = -1.

-1 | 1 -3 9 -5
| -1 4 -13
--------------
1 -4 13 -18

The remainder is -18, which means x = -1 is not a factor.

Next, we test x = 5.

5 | 1 -3 9 -5
| 5 10 95
-------------
1 2 19 90

The remainder is 90, which means x = 5 is not a factor.

Finally, we test x = -5.

-5 | 1 -3 9 -5
| -5 40 -245
--------------
1 -8 49 -250

The remainder is -250, which means x = -5 is not a factor.

Step 3: Use the factor theorem to determine the factor
Since none of the possible factors resulted in a remainder of zero, we can conclude that the given expression does not have any linear factors.

Step 4: Factorising the expression using the synthetic division results
From the synthetic division results, we obtain the expression: x^3 - 3x^2 + 9x - 5 = (x - 8)(x^2 + 19x + 49)

Therefore, the factorised form of the given expression is (x - 8)(x^2 + 19x + 49).

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