Factoring the expression x³ - 23x² + 142x - 120

To factor the given expression x³ - 23x² + 142x - 120, we can use a method called factoring by grouping. This method involves grouping terms together in pairs and then factoring out a common factor from each pair.

Step 1: Grouping the terms
We start by grouping the terms in pairs:
(x³ - 23x²) + (142x - 120)

Step 2: Factoring out the common factor
From the first pair of terms, we can factor out x²:
x²(x - 23)

From the second pair of terms, we can factor out 2:
2(71x - 60)

Now, we have the expression factored as:
x²(x - 23) + 2(71x - 60)

Step 3: Further simplification
We can simplify the expression by factoring out a common factor from the two terms:
x²(x - 23) + 2(71x - 60)
x²(x - 23) + 2(71x - 60)
x²(x - 23) + 2(71x - 60)

Now, we have the fully factored expression:
(x - 23)(x² + 2(71x - 60))

Step 4: Simplifying the remaining expression
To simplify the expression further, we can expand the second term:
(x - 23)(x² + 142x - 120)

Now, we have the completely factored expression:
(x - 23)(x² + 142x - 120)

In summary, the expression x³ - 23x² + 142x - 120 can be factored as (x - 23)(x² + 142x - 120).

(x+1) (x-10) (x-12)