Sure! Let's find the Highest Common Factor (HCF) of the given pairs of polynomials step by step.
### Part (a)
We need to find the HCF of the polynomials [tex]\(1 + 4x + 4x^2 - 16x^4\)[/tex] and [tex]\(1 + 2x - 8x^3 - 16x^4\)[/tex].
First, we write down the polynomials:
[tex]\[ p(x) = 1 + 4x + 4x^2 - 16x^4 \][/tex]
[tex]\[ q(x) = 1 + 2x - 8x^3 - 16x^4 \][/tex]
The HCF (or Greatest Common Divisor, GCD) of these two polynomials can be determined and is:
[tex]\[ \text{HCF}(p(x), q(x)) = 4x^2 + 2x + 1 \][/tex]
This is the polynomial that divides both [tex]\( p(x) \)[/tex] and [tex]\( q(x) \)[/tex] without leaving a remainder.
### Part (b)
Now, we need to find the HCF of the polynomials [tex]\(a^4 + a^2b^2 + b^4\)[/tex] and [tex]\(a^3 + 2a^2b + 2ab^2 + b^3\)[/tex].
First, we write down the polynomials:
[tex]\[ p(a, b) = a^4 + a^2b^2 + b^4 \][/tex]
[tex]\[ q(a, b) = a^3 + 2a^2b + 2ab^2 + b^3 \][/tex]
The HCF (or Greatest Common Divisor, GCD) of these two polynomials can be determined and is:
[tex]\[ \text{HCF}(p(a, b), q(a, b)) = a^2 + ab + b^2 \][/tex]
This is the polynomial that divides both [tex]\( p(a, b) \)[/tex] and [tex]\( q(a, b) \)[/tex] without leaving a remainder.
### Summary
- For part (a), the HCF is [tex]\( 4x^2 + 2x + 1 \)[/tex].
- For part (b), the HCF is [tex]\( a^2 + ab + b^2 \)[/tex].
