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Answer :
To find which polynomial factors as a perfect power (that is, it can be written as a constant multiplied by a binomial raised to an integer power), we can factor each option individually.
1. For the polynomial
[tex]$$25x^2 + 75x + 225,$$[/tex]
factoring gives
[tex]$$25(x^2 + 3x + 9).$$[/tex]
The quadratic factor does not have the form [tex]$(x + b)^n$[/tex], so it does not represent a perfect power derived from the Binomial Theorem.
2. For the polynomial
[tex]$$25x^2 + 300x + 225,$$[/tex]
factoring gives
[tex]$$25(x^2 + 12x + 9).$$[/tex]
Again, the quadratic is not a perfect square of a binomial.
3. For the polynomial
[tex]$$625x^4 + 1875x^3 + 5625x^2 + 16875x + 50625,$$[/tex]
factoring gives
[tex]$$625(x^4 + 3x^3 + 9x^2 + 27x + 81).$$[/tex]
The quartic inside does not factor as a single binomial raised to a power.
4. For the polynomial
[tex]$$625x^4 + 7500x^3 + 33750x^2 + 67500x + 50625,$$[/tex]
factoring gives
[tex]$$625(x+3)^4.$$[/tex]
Here, the expression is exactly in the form [tex]$$a\,(x+b)^n,$$[/tex] with [tex]$a=625$[/tex], [tex]$b=3$[/tex], and [tex]$n=4$[/tex]. This is the binomial expansion of [tex]$(x+3)^4$[/tex] multiplied by 625.
Since the only option that factors into a constant times a perfect power (a binomial raised to a power) is option 4, it can be factored using the Binomial Theorem.
Thus, the answer is option 4.
1. For the polynomial
[tex]$$25x^2 + 75x + 225,$$[/tex]
factoring gives
[tex]$$25(x^2 + 3x + 9).$$[/tex]
The quadratic factor does not have the form [tex]$(x + b)^n$[/tex], so it does not represent a perfect power derived from the Binomial Theorem.
2. For the polynomial
[tex]$$25x^2 + 300x + 225,$$[/tex]
factoring gives
[tex]$$25(x^2 + 12x + 9).$$[/tex]
Again, the quadratic is not a perfect square of a binomial.
3. For the polynomial
[tex]$$625x^4 + 1875x^3 + 5625x^2 + 16875x + 50625,$$[/tex]
factoring gives
[tex]$$625(x^4 + 3x^3 + 9x^2 + 27x + 81).$$[/tex]
The quartic inside does not factor as a single binomial raised to a power.
4. For the polynomial
[tex]$$625x^4 + 7500x^3 + 33750x^2 + 67500x + 50625,$$[/tex]
factoring gives
[tex]$$625(x+3)^4.$$[/tex]
Here, the expression is exactly in the form [tex]$$a\,(x+b)^n,$$[/tex] with [tex]$a=625$[/tex], [tex]$b=3$[/tex], and [tex]$n=4$[/tex]. This is the binomial expansion of [tex]$(x+3)^4$[/tex] multiplied by 625.
Since the only option that factors into a constant times a perfect power (a binomial raised to a power) is option 4, it can be factored using the Binomial Theorem.
Thus, the answer is option 4.
Thank you for reading the article Which polynomial can be factored using the binomial theorem A tex 25x 2 75x 225 tex B tex 25x 2 300x 225 tex C tex. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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