To find the Highest Common Factor (H.C.F.), also known as the Greatest Common Divisor (GCD), of the two given polynomials [tex]\( 8a^3 + b^3 \)[/tex] and [tex]\( 16a^4 + 4a b + b \)[/tex], we undertake the following steps:
1. Express the Polynomials:
- Let [tex]\( \text{poly1} = 8a^3 + b^3 \)[/tex]
- Let [tex]\( \text{poly2} = 16a^4 + 4a b + b \)[/tex]
2. Factorize the Polynomials:
- [tex]\( \text{poly1} = 8a^3 + b^3 \)[/tex] looks for factorization as a sum of cubes:
[tex]\( 8a^3 = (2a)^3 \)[/tex],
[tex]\( b^3 = (b)^3 \)[/tex],
Thus, [tex]\( 8a^3 + b^3 = (2a + b)((2a)^2 - (2a)(b) + b^2) = (2a + b)(4a^2 - 2ab + b^2) \)[/tex].
- For [tex]\( \text{poly2} = 16a^4 + 4ab + b \)[/tex], it is not as straightforward for standard factorization.
3. Identify Common Factors:
- Upon looking at factorized poly1, [tex]\( (2a + b) \)[/tex] is straightforward as a potential common factor.
- However, after a detailed examination of poly2, we identify no significant polynomial terms that precisely fit or factor cleanly into [tex]\( 16a^4 + 4ab + b \)[/tex].
4. Evaluate H.C.F.:
- Since we cannot find any common terms upon further inspection, between the factorized result of poly1 and poly2, the polynomials share no common factor besides the trivial factor [tex]\(1\)[/tex].
Therefore, the H.C.F. (Greatest Common Divisor) of [tex]\( 8a^3 + b^3 \)[/tex] and [tex]\( 16a^4 + 4ab + b \)[/tex] is:
[tex]\[ \boxed{1} \][/tex]
This means the two polynomials are co-prime, having no non-trivial common factors.
