Let's find the L.C.M. of 2(x – 4) and (x² – 3x + 2)

Let us understand what L.C.M, Least Common Multiple is.

Least common multiple of two or more integers, is the smallest positive integer that is divisible by these two or more integers.

In order to find L.C.M, we need to find factors of each terms.

Let us factorize 2(x – 4).

2(x – 4) = 2 × (x – 4)

Let us factorize (x² – 3x + 2).

x² – 3x + 2 = x² – (2x + x) + 2

[∵, Sum of -2 and -1 is -3, and its multiplication is 2]

⇒ x² – 3x + 2 = x² – 2x – x + 2

⇒ x² – 3x + 2 = x(x – 2) – (x – 2)

[∵, common from first two terms is x and last two terms is -1]

⇒ x² – 3x + 2 = (x – 2)(x – 1)

[∵, common from the two terms is (x – 2)]

Now, we need to find the factor of highest power.

Factors of 2(x – 4) = 2, (x – 4)

Factors of x² – 3x + 2 = (x – 2), (x – 1)

The factors are 2, (x – 4), (x – 2) and (x – 1).

Multiplying these factors, we get

2 × (x – 4) × (x – 2) × (x – 1) = 2(x – 4)(x – 2)(x – 1)

Thus, the lcm of 2(x – 4) and (x² – 3x + 2) is 2(x – 4)(x – 2)(x – 1).