Let's analyze each polynomial to see which one can be factored using the binomial theorem:
1. Polynomial 1: [tex]\(64 x^3 + 48 x^2 + 36 x + 27\)[/tex]
This polynomial can be rewritten and factored as:
[tex]\[
64 x^3 + 48 x^2 + 36 x + 27 = (4x + 3)(16x^2 + 9)
\][/tex]
2. Polynomial 2: [tex]\(64 x^3 + 96 x^2 + 72 x + 27\)[/tex]
This polynomial can be rewritten and factored as:
[tex]\[
64 x^3 + 96 x^2 + 72 x + 27 = (4x + 3)(16x^2 + 12x + 9)
\][/tex]
3. Polynomial 3: [tex]\(256 x^4 - 192 x^3 + 144 x^2 - 108 x + 81\)[/tex]
This polynomial doesn't factor nicely using the binomial theorem and stays as:
[tex]\[
256 x^4 - 192 x^3 + 144 x^2 - 108 x + 81
\][/tex]
4. Polynomial 4: [tex]\(256 x^4 - 768 x^3 + 864 x^2 - 432 x + 81\)[/tex]
This polynomial can be rewritten and factored using the binomial theorem as:
[tex]\[
256 x^4 - 768 x^3 + 864 x^2 - 432 x + 81 = (4x - 3)^4
\][/tex]
From our analysis, Polynomial 4: [tex]\(256 x^4 - 768 x^3 + 864 x^2 - 432 x + 81\)[/tex] can be factored using the binomial theorem as [tex]\((4x - 3)^4\)[/tex].
Therefore, the polynomial that can be factored using the binomial theorem is:
[tex]\[
256 x^4 - 768 x^3 + 864 x^2 - 432 x + 81
\][/tex]
