Certainly! Let's go through the steps to solve the problem step-by-step.
### 1.1 Write down all the possible factors for each of the following expressions.
#### Expression 1: \(\( 3a^2b \)\)
To factorize \(\( 3a^2b \)\), we need to break it down into its constituent parts.
- The coefficient 3 is already a prime number, so it factors as \(\( 3 \)\).
- \(\( a^2 \)\) indicates that \(\( a \)\) is raised to the power of 2, meaning \(\( a \cdot a \)\).
- \(\( b \)\) is a single variable.
Therefore, the factors of \(\( 3a^2b \)\) are:
\(\[ 3 \cdot a \cdot a \cdot b \]\)
This can also be written as:
\(\[ 3 \cdot a^2 \cdot b \]\)
#### Expression 2: \(\( 4ab^2 \)\)
To factorize \(\( 4ab^2 \)\), we need to break it down into its constituent parts.
- The coefficient 4 can be factorized into primes as \(\( 2 \cdot 2 \)\).
- \(\( a \)\) is a single variable.
- \(\( b^2 \)\) indicates that \(\( b \)\) is raised to the power of 2, meaning \(\( b \cdot b \)\).
Therefore, the factors of \(\( 4ab^2 \)\) are:
\(\[ 4 \cdot a \cdot b \cdot b \]\)
This can also be written as:
\(\[ 4 \cdot a \cdot b^2 \]\)
### 1.2 Determine the Highest Common Factor (H.C.F.) of the algebraic expressions.
To find the H.C.F. (Highest Common Factor) of \(\( 3a^2b \)\) and \(\( 4ab^2 \)\), we look for the highest common factors in both the numerical coefficients and the variable terms.
#### Numerical Coefficient:
- The numerical coefficients of the given expressions are 3 and 4.
- The highest common factor (H.C.F.) of 3 and 4 is 1, since 3 and 4 are coprime (they have no common factors other than 1).
#### Variable Factors:
- For \(\( a \)\):
- \(\( 3a^2b \)\) has \(\( a^2 \)\).
- \(\( 4ab^2 \)\) has \(\( a \)\).
- The H.C.F. is \(\( a \)\), as \(\( a \)\) is the lowest power common to both.
- For \(\( b \)\):
- \(\( 3a^2b \)\) has \(\( b \)\).
- \(\( 4ab^2 \)\) has \(\( b^2 \)\).
- The H.C.F. is \(\( b \)\), as \(\( b \)\) is the lowest power common to both.
Putting these together, the H.C.F. of the algebraic expressions \(\( 3a^2b \)\) and \(\( 4ab^2 \)\) is:
\(\[ a \cdot b = ab \]\)
So, the Highest Common Factor (H.C.F.) of \(\( 3a^2b \)\) and \(\( 4ab^2 \)\) is:
\(\[ ab \]\)
### Summary
1.1. Factors:
- \(\( 3a^2b = 3 \cdot a^2 \cdot b \)\)
- \(\( 4ab^2 = 4 \cdot a \cdot b^2 \)\)
1.2. The H.C.F. of the algebraic expressions \(\( 3a^2b \)\) and \(\( 4ab^2 \)\) is:
\(\[ ab \]\)
