The only polynomial that can be factored using the binomial theorem is; D: 625x⁴ + 7500x³ + 33750x² + 67500x + 50,625
How to use the binomial Theorem?
1) 25x² + 75x + 225
Let us factorize 25 out to get;
25(x² + 3x + 9)
This cannot be factored further to a perfect square.
2) 25x² + 300x + 225
Let us factorize 25 out to get;
25(x² + 12x + 9)
This cannot be factored further to a perfect square.
3) 625x⁴ + 1,875x³ + 5,625x² + 16,875x + 50,625
Let us factorize 25 out to get;
625(x⁴ + 3x³ + 9x² + 27x + 81)
This can be factored into a perfect square as;
625(x + 3)⁴
Using Pascals triangle in binomial theorem gives;
625(1(x⁴ * 3⁰) + 4(x³ * 3¹) + 6(x² * 3²) + 4(x¹ * 3³) + 1(3⁴))
= 625(x⁴ + 12x³ + 54x² + 108x + 81)
This does not tally with the given expression
4) 625x⁴ + 7500x³ + 33750x² + 67500x + 50,625
Let us factorize 25 out to get;
625(x⁴ + 12x³ + 54x² + 108x + 81)
This can be factored into a perfect square as;
625(x + 3)⁴
Using Pascals triangle in binomial theorem gives;
625(1(x⁴ * 3⁰) + 4(x³ * 3¹) + 6(x² * 3²) + 4(x¹ * 3³) + 1(3⁴))
= 625(x⁴ + 12x³ + 54x² + 108x + 81)
Read more about Binomial Theorem at; https://brainly.com/question/13602562
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