Factorize the given expression using the factor theorem. The expression is:
x^3 - 23x^2 + 142x - 120

To factorize the given expression using the factor theorem, we proceed step by step.

Given expression:
x³ - 23x² + 142x - 120

Step 1: Apply the Factor Theorem
The factor theorem states that if (x - a) is a factor of the polynomial, then the value of the polynomial at x = a will be 0, i.e., P(a) = 0.

We start by testing possible rational roots using the rational root theorem, which suggests that the possible roots are factors of the constant term (-120) divided by factors of the leading coefficient (1).

Factors of -120: ±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±20, ±24, ±30, ±40, ±60, ±120

Factors of 1: ±1

Thus, the possible rational roots are: ±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±20, ±24, ±30, ±40, ±60, ±120

Step 2: Test Possible Roots
Start testing with some values for x. We test x = 2 first.

Substitute x = 2 into the polynomial:

P(2) = 2³ - 23(2²) + 142(2) - 120
= 8 - 23(4) + 142(2) - 120
= 8 - 92 + 284 - 120
= 8 + 284 - 92 - 120
= 0

Since P(2) = 0, we know that (x - 2) is a factor of the polynomial.

Step 3: Divide the Polynomial by (x - 2)
Now, divide the given polynomial by (x - 2) using synthetic division or long division.

Synthetic Division:
We divide the polynomial x³ - 23x² + 142x - 120 by (x - 2).

Set up the coefficients: 1, -23, 142, -120
Use 2 for synthetic division:
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2 | 1 -23 142 -120
| 2 -42 200
------------------------
1 -21 100 0
The quotient is x² - 21x + 60, and the remainder is 0.

Step 4: Factor the Quotient
Now, factor the quadratic expression x² - 21x + 60.

We need two numbers that multiply to 60 and add up to -21. These numbers are -15 and -6.

Thus, we can factor x² - 21x + 60 as:

x² - 21x + 60 = (x - 15)(x - 6)

Step 5: Write the Final Factorization
Now, combine all the factors:

x³ - 23x² + 142x - 120 = (x - 2)(x - 15)(x - 6)

Thus, the factorization of the given expression is:

(x - 2)(x - 15)(x - 6)