Write down all the possible factors for each of the following expressions. Determine the H.C.F. of the algebraic expression.

To find all factors, we consider the factors of the numerical coefficient and each variable raised to its power.
Factors of 3: $1, 3$
Factors of $a^2$: $1, a, a^2$
Factors of $b$: $1, b$
Multiplying all combinations of these factors gives:

$$1, 3, a, a^2, b, 3a, 3a^2, 3b, ab, a^2b, 3ab, 3a^2b$$

Factors of 4: $1, 2, 4$
Factors of $a$: $1, a$
Factors of $b^2$: $1, b, b^2$
Multiplying all combinations of these factors gives:

$$1, 2, 4, a, b, b^2, 2a, 2b, 2b^2, 4a, 4b, 4b^2, ab, ab^2, 2ab, 2ab^2, 4ab, 4ab^2$$

of the algebraic expression $3a^2b + 4ab^2$.
The H.C.F. of the expression is the H.C.F. of its terms, $3a^2b$ and $4ab^2$.
First, find the H.C.F. of the numerical coefficients:
H.C.F. of 3 and 4 is 1.
Next, find the H.C.F. of the variable parts:
For $a$: The lowest power of $a$ in $a^2$ and $a$ is $a^1$, which is $a$.
For $b$: The lowest power of $b$ in $b$ and $b^2$ is $b^1$, which is $b$.
Multiply the common numerical and variable factors:

$$\text{H.C.F.} = 1 \times a \times b = ab$$

The H.C.F. of the algebraic expression is $\text{ab}$.