Answer :
1.1 Writing down all the possible factors for each of the following expressions:
For \( 3a^2b \):
- Numerical factors: 1, 3
- Algebraic factors: \(a\), \(a^2\), \(b\), \(ab\), \(a^2b\)
The factors of \(3a^2b\) are the product of its numerical factor (which is 3) and its algebraic factors (which are \(a\), \(a^2\), \(b\), \(ab\), \(a^2b\)). It's important to note that \(a\) and \(a^2\) are different factors, where \(a^2\) is \(a\) multiplied by itself.
For \( 4ab^2 \):
- Numerical factors: 1, 2, 4
- Algebraic factors: \(a\), \(b\), \(b^2\), \(ab\), \(ab^2\)
The factors of \(4ab^2\) are the product of its numerical factors (which are 2 and 4, since 2 multiplied by itself is 4) and its algebraic factors (which are \(a\), \(b\), \(b^2\), \(ab\), \(ab^2\)).
Just like before, \(b\) and \(b^2\) are different factors, where \(b^2\) is \(b\) multiplied by itself.
1.2 Determining the H.C.F. (highest common factor) of the algebraic expression:
To find the H.C.F of \(3a^2b\) and \(4ab^2\), we need to determine the highest common factor that each term shares. This means we are looking for the largest combination of numerical and algebraic factors that both expressions have in common.
Numerically, the highest common factor of 3 and 4 is 1, since there are no common divisors other than 1.
Algebraically, we look at the factors of \(a\) and \(b\) in each term:
- Both terms have at least one \(a\) in common.
- Both terms have at least one \(b\) in common.
Considering the powers of each variable, we can only take the lowest power that appears in both terms, otherwise, it wouldn't be common to them. This means we take \(a^1\) (or simply \(a\)) and \(b^1\) (or simply \(b\)) as the variables can only be considered with their lowest common exponents.
Therefore, the H.C.F of the algebraic expression \(3a^2b\) and \(4ab^2\) would be \(ab\).
