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Answer :
Certainly! Let's examine each polynomial to determine if we can rewrite them in a squared form.
### Polynomial 18: [tex]\( x^6 + 8x^3 + 3 \)[/tex]
To rewrite a polynomial in a squared form, we're looking for something like [tex]\((ax^m + b)^2\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(m\)[/tex] are numbers such that expanding [tex]\((ax^m + b)^2\)[/tex] would result in terms like [tex]\(x^6\)[/tex], [tex]\(x^3\)[/tex], and constants.
However, this particular polynomial doesn't fit any recognizable pattern for completing the square or factoring into a perfect square. The middle term [tex]\(8x^3\)[/tex] suggests a linear term, but the powers [tex]\(x^6\)[/tex] and constant [tex]\(3\)[/tex] don't allow for a typical square relationship.
Therefore, we conclude that this polynomial cannot be rewritten into a squared form.
### Polynomial 19: [tex]\( x^{14} - 6x^7 + 7 \)[/tex]
Similarly, looking at this polynomial, we have terms [tex]\(x^{14}\)[/tex], [tex]\(-6x^7\)[/tex], and [tex]\(7\)[/tex]. The idea is to express it in the form of [tex]\((ax^m + b)^2\)[/tex], but the structure doesn't match a recognizable pattern for completing the square or square factorization due to the combination of powers and coefficients.
Therefore, this polynomial also cannot be rewritten into a squared form.
### Polynomial 20: [tex]\( x^6 + 8x^3 + 3 \)[/tex]
This is identical to polynomial 18. As we have already determined, it cannot be rewritten into a squared form for the same reasons mentioned above.
In summary, none of the given polynomials can be rewritten into a squared form as they do not match the patterns needed for such a transformation.
### Polynomial 18: [tex]\( x^6 + 8x^3 + 3 \)[/tex]
To rewrite a polynomial in a squared form, we're looking for something like [tex]\((ax^m + b)^2\)[/tex], where [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(m\)[/tex] are numbers such that expanding [tex]\((ax^m + b)^2\)[/tex] would result in terms like [tex]\(x^6\)[/tex], [tex]\(x^3\)[/tex], and constants.
However, this particular polynomial doesn't fit any recognizable pattern for completing the square or factoring into a perfect square. The middle term [tex]\(8x^3\)[/tex] suggests a linear term, but the powers [tex]\(x^6\)[/tex] and constant [tex]\(3\)[/tex] don't allow for a typical square relationship.
Therefore, we conclude that this polynomial cannot be rewritten into a squared form.
### Polynomial 19: [tex]\( x^{14} - 6x^7 + 7 \)[/tex]
Similarly, looking at this polynomial, we have terms [tex]\(x^{14}\)[/tex], [tex]\(-6x^7\)[/tex], and [tex]\(7\)[/tex]. The idea is to express it in the form of [tex]\((ax^m + b)^2\)[/tex], but the structure doesn't match a recognizable pattern for completing the square or square factorization due to the combination of powers and coefficients.
Therefore, this polynomial also cannot be rewritten into a squared form.
### Polynomial 20: [tex]\( x^6 + 8x^3 + 3 \)[/tex]
This is identical to polynomial 18. As we have already determined, it cannot be rewritten into a squared form for the same reasons mentioned above.
In summary, none of the given polynomials can be rewritten into a squared form as they do not match the patterns needed for such a transformation.
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