Which polynomial can be factored using the binomial theorem?

A. [tex]64x^3 + 48x^2 + 36x + 27[/tex]

B. [tex]64x^3 + 96x^2 + 72x + 27[/tex]

C. [tex]256x^4 - 192x^3 + 144x^2 - 108x + 81[/tex]

D. [tex]256x^4 - 768x^3 + 864x^2 - 432x + 81[/tex]



Answer :

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]