To determine the Highest Common Factor (HCF) of the expressions [tex]\(9(x + y)\)[/tex] and [tex]\(3(x^2 - y^2)\)[/tex], let's follow these steps:
1. Factorize Each Expression:
- Expression 1: [tex]\(9(x + y)\)[/tex]
This expression is already in its simplest factorized form.
- Expression 2: [tex]\(3(x^2 - y^2)\)[/tex]
Notice that [tex]\(x^2 - y^2\)[/tex] is a difference of squares, which can be factored as [tex]\((x - y)(x + y)\)[/tex]. Therefore:
[tex]\[
3(x^2 - y^2) = 3(x - y)(x + y)
\][/tex]
2. Identify Common Factors:
- From the factorized forms, we have:
[tex]\[
9(x + y) = 9 \cdot (x + y)
\][/tex]
[tex]\[
3(x^2 - y^2) = 3 \cdot (x - y) \cdot (x + y)
\][/tex]
- We see that both expressions share a common factor of [tex]\((x + y)\)[/tex] and a numerical factor related to their coefficients.
3. Determine Numerical HCF of Coefficients:
- The coefficients in the expressions are 9 and 3.
- The HCF of 9 and 3 is 3.
4. Combine the Common Factors:
- Combine the numerical HCF with the common algebraic factor:
[tex]\[
3 \cdot (x + y)
\][/tex]
Therefore, the HCF of [tex]\(9(x+y)\)[/tex] and [tex]\(3(x^2 - y^2)\)[/tex] is:
[tex]\[
3(x + y)
\][/tex]
