To determine the Highest Common Factor (H.C.F.) of the polynomials [tex]\( 1 + 4x + 4x^2 - 16x^4 \)[/tex] and [tex]\( 1 + 2x - 8x^3 - 16x^4 \)[/tex], follow these steps:
1. Identify the given polynomials:
We are provided with two polynomials:
[tex]\[
P(x) = 1 + 4x + 4x^2 - 16x^4
\][/tex]
[tex]\[
Q(x) = 1 + 2x - 8x^3 - 16x^4
\][/tex]
2. Factorizing the polynomials:
To find the H.C.F., we must understand how to factorize these polynomials. Factorizing helps in identifying common factors between the two polynomials.
3. Find common factors:
Upon inspecting both polynomials, identify common terms or factors that can be pulled out.
4. Calculate the H.C.F.:
Through either manual factorization or mathematical tools, deduce the greatest polynomial that divides both [tex]\( P(x) \)[/tex] and [tex]\( Q(x) \)[/tex] without leaving a remainder.
5. Result:
After analysis, you find that the H.C.F. of the given polynomials [tex]\( 1 + 4x + 4x^2 - 16x^4 \)[/tex] and [tex]\( 1 + 2x - 8x^3 - 16x^4 \)[/tex] is:
[tex]\[
4x^2 + 2x + 1
\][/tex]
This H.C.F. represents the highest polynomial that can exactly divide both given polynomials. Thus, the H.C.F. of the polynomials [tex]\( 1 + 4x + 4x^2 - 16x^4 \)[/tex] and [tex]\( 1 + 2x - 8x^3 - 16x^4 \)[/tex] is [tex]\( 4x^2 + 2x + 1 \)[/tex].
