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: [tex]\( 3a^2b \)[/tex]
To factorize [tex]\( 3a^2b \)[/tex], we need to break it down into its constituent parts.
- The coefficient 3 is already a prime number, so it factors as [tex]\( 3 \)[/tex].
- [tex]\( a^2 \)[/tex] indicates that [tex]\( a \)[/tex] is raised to the power of 2, meaning [tex]\( a \cdot a \)[/tex].
- [tex]\( b \)[/tex] is a single variable.
Therefore, the factors of [tex]\( 3a^2b \)[/tex] are:
[tex]\[ 3 \cdot a \cdot a \cdot b \][/tex]
This can also be written as:
[tex]\[ 3 \cdot a^2 \cdot b \][/tex]
#### Expression 2: [tex]\( 4ab^2 \)[/tex]
To factorize [tex]\( 4ab^2 \)[/tex], we need to break it down into its constituent parts.
- The coefficient 4 can be factorized into primes as [tex]\( 2 \cdot 2 \)[/tex].
- [tex]\( a \)[/tex] is a single variable.
- [tex]\( b^2 \)[/tex] indicates that [tex]\( b \)[/tex] is raised to the power of 2, meaning [tex]\( b \cdot b \)[/tex].
Therefore, the factors of [tex]\( 4ab^2 \)[/tex] are:
[tex]\[ 4 \cdot a \cdot b \cdot b \][/tex]
This can also be written as:
[tex]\[ 4 \cdot a \cdot b^2 \][/tex]
### 1.2 Determine the Highest Common Factor (H.C.F.) of the algebraic expressions.
To find the H.C.F. (Highest Common Factor) of [tex]\( 3a^2b \)[/tex] and [tex]\( 4ab^2 \)[/tex], 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 [tex]\( a \)[/tex]:
- [tex]\( 3a^2b \)[/tex] has [tex]\( a^2 \)[/tex].
- [tex]\( 4ab^2 \)[/tex] has [tex]\( a \)[/tex].
- The H.C.F. is [tex]\( a \)[/tex], as [tex]\( a \)[/tex] is the lowest power common to both.
- For [tex]\( b \)[/tex]:
- [tex]\( 3a^2b \)[/tex] has [tex]\( b \)[/tex].
- [tex]\( 4ab^2 \)[/tex] has [tex]\( b^2 \)[/tex].
- The H.C.F. is [tex]\( b \)[/tex], as [tex]\( b \)[/tex] is the lowest power common to both.
Putting these together, the H.C.F. of the algebraic expressions [tex]\( 3a^2b \)[/tex] and [tex]\( 4ab^2 \)[/tex] is:
[tex]\[ a \cdot b = ab \][/tex]
So, the Highest Common Factor (H.C.F.) of [tex]\( 3a^2b \)[/tex] and [tex]\( 4ab^2 \)[/tex] is:
[tex]\[ ab \][/tex]
### Summary
1.1. Factors:
- [tex]\( 3a^2b = 3 \cdot a^2 \cdot b \)[/tex]
- [tex]\( 4ab^2 = 4 \cdot a \cdot b^2 \)[/tex]
1.2. The H.C.F. of the algebraic expressions [tex]\( 3a^2b \)[/tex] and [tex]\( 4ab^2 \)[/tex] is:
[tex]\[ ab \][/tex]
