Factorising x^3-23x^2+142x-120

To factorise the given expression x^3-23x^2+142x-120, we can use a method called grouping. This involves grouping the terms in pairs and factoring out the common factor from each pair.

Grouping the terms:
x^3-23x^2+142x-120
= (x^3-23x^2) + (142x-120)

Now, let's factor out the common factor from each pair.

Factorising the first pair:
x^3-23x^2
= x^2(x-23)

Factorising the second pair:
142x-120
= 2(71x-60)

Now, we have factored out the common factors from each pair.

Combining the factorised pairs:
(x^2(x-23)) + (2(71x-60))

Now, we can see that both terms have a common factor of (x-23). Let's factor out this common factor.

Factoring out the common factor:
(x^2(x-23)) + (2(71x-60))
= (x-23)(x^2+2(71x-60))

Finally, we have fully factorised the expression x^3-23x^2+142x-120 as (x-23)(x^2+2(71x-60)).

Summary:
To factorise x^3-23x^2+142x-120, we used the grouping method. We grouped the terms in pairs and factored out the common factor from each pair. Then, we combined the factorised pairs and factored out the common factor (x-23) to obtain the final factorised form: (x-23)(x^2+2(71x-60)).