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Answer :
Certainly! Let's analyze Cecile's work to check if she factored the polynomial correctly.
1. Initial Polynomial: Cecile is working with the expression [tex]\(16x^5 - 9\)[/tex].
2. Cecile's Format: She begins by writing it as [tex]\(16x^6 + 0x - 9\)[/tex]. This is a mistake because the original polynomial is [tex]\(16x^5 - 9\)[/tex], not [tex]\(16x^6 - 9\)[/tex]. The powers and expression do not match.
3. Application of the X Method: Cecile's attempt to use the X Method also has issues:
- Typically, the X Method involves finding two numbers that multiply to the product of the coefficient of the first term and the constant term in the polynomial. However, Cecile's X Method application does not clearly follow this, and adding [tex]\(0x\)[/tex] doesn't contribute correctly to factoring the expression.
4. Factoring Attempt: She then writes:
[tex]\[
16x^6 + 12x^3 - 12x^3 - 9 = 4x^3(4x^3 + 3) + (-3)(4x^3 - 3)
\][/tex]
Here, she incorrectly splits the expression into terms involving [tex]\(x^3\)[/tex], which aren't aligned with the original expression [tex]\(16x^5 - 9\)[/tex].
5. Final Factored Form: Cecile concludes with:
[tex]\[
(4x^3 + 3)(4x^3 - 3)
\][/tex]
Again, this reflects a mistake because these factors result from an incorrect expression initially written as [tex]\(16x^6\)[/tex].
6. Conclusion: Because Cecile started with an incorrect expression, her factoring steps are ultimately incorrect for the polynomial [tex]\(16x^5 - 9\)[/tex].
Thus, the correct assessment is that Cecile did not factor the polynomial correctly. Specifically, the polynomial [tex]\(16x^6 + 12x^3 - 12x^3 - 9\)[/tex] is not equivalent to [tex]\(16x^5 - 9\)[/tex].
1. Initial Polynomial: Cecile is working with the expression [tex]\(16x^5 - 9\)[/tex].
2. Cecile's Format: She begins by writing it as [tex]\(16x^6 + 0x - 9\)[/tex]. This is a mistake because the original polynomial is [tex]\(16x^5 - 9\)[/tex], not [tex]\(16x^6 - 9\)[/tex]. The powers and expression do not match.
3. Application of the X Method: Cecile's attempt to use the X Method also has issues:
- Typically, the X Method involves finding two numbers that multiply to the product of the coefficient of the first term and the constant term in the polynomial. However, Cecile's X Method application does not clearly follow this, and adding [tex]\(0x\)[/tex] doesn't contribute correctly to factoring the expression.
4. Factoring Attempt: She then writes:
[tex]\[
16x^6 + 12x^3 - 12x^3 - 9 = 4x^3(4x^3 + 3) + (-3)(4x^3 - 3)
\][/tex]
Here, she incorrectly splits the expression into terms involving [tex]\(x^3\)[/tex], which aren't aligned with the original expression [tex]\(16x^5 - 9\)[/tex].
5. Final Factored Form: Cecile concludes with:
[tex]\[
(4x^3 + 3)(4x^3 - 3)
\][/tex]
Again, this reflects a mistake because these factors result from an incorrect expression initially written as [tex]\(16x^6\)[/tex].
6. Conclusion: Because Cecile started with an incorrect expression, her factoring steps are ultimately incorrect for the polynomial [tex]\(16x^5 - 9\)[/tex].
Thus, the correct assessment is that Cecile did not factor the polynomial correctly. Specifically, the polynomial [tex]\(16x^6 + 12x^3 - 12x^3 - 9\)[/tex] is not equivalent to [tex]\(16x^5 - 9\)[/tex].
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