Let's find the Highest Common Factor (H.C.F.) for each pair of expressions.
### a) [tex]\(a^2 - b^2\)[/tex] and [tex]\(a^3 + b^3\)[/tex]
1. Factorize the expressions:
- [tex]\(a^2 - b^2 = (a - b)(a + b)\)[/tex]
- [tex]\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)[/tex]
2. Find the common factors:
- The common factor between the factorizations is [tex]\(a + b\)[/tex].
So, the H.C.F. of [tex]\(a^2 - b^2\)[/tex] and [tex]\(a^3 + b^3\)[/tex] is:
[tex]\[a + b\][/tex]
### c) [tex]\(x^2 - 9\)[/tex] and [tex]\(3x + 9\)[/tex]
1. Factorize the expressions:
- [tex]\(x^2 - 9 = (x - 3)(x + 3)\)[/tex]
- [tex]\(3x + 9 = 3(x + 3)\)[/tex]
2. Find the common factors:
- The common factor between the factorizations is [tex]\(x + 3\)[/tex].
So, the H.C.F. of [tex]\(x^2 - 9\)[/tex] and [tex]\(3x + 9\)[/tex] is:
[tex]\[x + 3\][/tex]
### e) [tex]\(4x^2 - 100\)[/tex] and [tex]\(4x + 20\)[/tex]
1. Factorize the expressions:
- [tex]\(4x^2 - 100 = 4(x^2 - 25) = 4(x - 5)(x + 5)\)[/tex]
- [tex]\(4x + 20 = 4(x + 5)\)[/tex]
2. Find the common factors:
- The common factor between the factorizations is [tex]\(4(x + 5)\)[/tex].
So, the H.C.F. of [tex]\(4x^2 - 100\)[/tex] and [tex]\(4x + 20\)[/tex] is:
[tex]\[4(x + 5)\][/tex]
### Summary
- The H.C.F. of [tex]\(a^2 - b^2\)[/tex] and [tex]\(a^3 + b^3\)[/tex] is [tex]\(a + b\)[/tex].
- The H.C.F. of [tex]\(x^2 - 9\)[/tex] and [tex]\(3x + 9\)[/tex] is [tex]\(x + 3\)[/tex].
- The H.C.F. of [tex]\(4x^2 - 100\)[/tex] and [tex]\(4x + 20\)[/tex] is [tex]\(4(x + 5)\)[/tex].
