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Answer :
To determine which polynomial can be factored using the binomial theorem, we'll factor each polynomial step by step.
1. Polynomial: [tex]\( 25x^2 + 75x + 225 \)[/tex]
- Factor out the common factor from each term:
[tex]\[
25x^2 + 75x + 225 = 25(x^2 + 3x + 9)
\][/tex]
- The expression inside the parentheses, [tex]\( x^2 + 3x + 9 \)[/tex], does not fit the pattern of a binomial square, so it cannot be factored using the binomial theorem.
2. Polynomial: [tex]\( 25x^2 + 300x + 225 \)[/tex]
- Factor out the common factor from each term:
[tex]\[
25x^2 + 300x + 225 = 25(x^2 + 12x + 9)
\][/tex]
- The expression inside the parentheses, [tex]\( x^2 + 12x + 9 \)[/tex], doesn't precisely fit the binomial pattern such as [tex]\((a+b)^2\)[/tex].
3. Polynomial: [tex]\( 625x^4 + 1875x^3 + 5625x^2 + 16875x + 50625 \)[/tex]
- Factor out the common factor from each term:
[tex]\[
625(x^4 + 3x^3 + 9x^2 + 27x + 81)
\][/tex]
- Although we have factored out [tex]\( 625 \)[/tex], the polynomial inside, [tex]\( x^4 + 3x^3 + 9x^2 + 27x + 81 \)[/tex], is not a simple binomial expansion.
4. Polynomial: [tex]\( 625x^4 + 7500x^3 + 33750x^2 + 67500x + 50625 \)[/tex]
- Factor out the common factor from each term:
[tex]\[
625(x^4 + 12x^3 + 54x^2 + 108x + 81)
\][/tex]
- This polynomial can be factored further by recognizing patterns:
[tex]\[
625(x + 3)^4
\][/tex]
- The expression [tex]\( (x + 3)^4 \)[/tex] fits the binomial expansion theorem perfectly, i.e., it expands into [tex]\( x^4 + 4(x^3)(3) + 6(x^2)(3^2) + 4(x)(3^3) + 3^4 \)[/tex].
So, the polynomial that can be factored using the binomial theorem is:
[tex]\[ \boxed{625x^4 + 7500x^3 + 33750x^2 + 67500x + 50625} \][/tex]
It can be factored as:
[tex]\[ 625(x + 3)^4 \][/tex]
1. Polynomial: [tex]\( 25x^2 + 75x + 225 \)[/tex]
- Factor out the common factor from each term:
[tex]\[
25x^2 + 75x + 225 = 25(x^2 + 3x + 9)
\][/tex]
- The expression inside the parentheses, [tex]\( x^2 + 3x + 9 \)[/tex], does not fit the pattern of a binomial square, so it cannot be factored using the binomial theorem.
2. Polynomial: [tex]\( 25x^2 + 300x + 225 \)[/tex]
- Factor out the common factor from each term:
[tex]\[
25x^2 + 300x + 225 = 25(x^2 + 12x + 9)
\][/tex]
- The expression inside the parentheses, [tex]\( x^2 + 12x + 9 \)[/tex], doesn't precisely fit the binomial pattern such as [tex]\((a+b)^2\)[/tex].
3. Polynomial: [tex]\( 625x^4 + 1875x^3 + 5625x^2 + 16875x + 50625 \)[/tex]
- Factor out the common factor from each term:
[tex]\[
625(x^4 + 3x^3 + 9x^2 + 27x + 81)
\][/tex]
- Although we have factored out [tex]\( 625 \)[/tex], the polynomial inside, [tex]\( x^4 + 3x^3 + 9x^2 + 27x + 81 \)[/tex], is not a simple binomial expansion.
4. Polynomial: [tex]\( 625x^4 + 7500x^3 + 33750x^2 + 67500x + 50625 \)[/tex]
- Factor out the common factor from each term:
[tex]\[
625(x^4 + 12x^3 + 54x^2 + 108x + 81)
\][/tex]
- This polynomial can be factored further by recognizing patterns:
[tex]\[
625(x + 3)^4
\][/tex]
- The expression [tex]\( (x + 3)^4 \)[/tex] fits the binomial expansion theorem perfectly, i.e., it expands into [tex]\( x^4 + 4(x^3)(3) + 6(x^2)(3^2) + 4(x)(3^3) + 3^4 \)[/tex].
So, the polynomial that can be factored using the binomial theorem is:
[tex]\[ \boxed{625x^4 + 7500x^3 + 33750x^2 + 67500x + 50625} \][/tex]
It can be factored as:
[tex]\[ 625(x + 3)^4 \][/tex]
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