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Which polynomial can be factored using the binomial theorem?

1) [tex]25x^2 + 75x + 225[/tex]

2) [tex]25x^2 + 300x + 225[/tex]

3) [tex]625x^4 + 1,875x^3 + 5,625x^2 + 16,875x + 50,625[/tex]

4) [tex]625x^4 + 7,500x^3 + 33,750x^2 + 67,500x + 50,625[/tex]

Answer :

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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