Cecile used the X method to factor [tex]16x^5 - 9[/tex].

1. [tex]16x^6 + 0x - 9[/tex]
2. X method:
3. [tex]16x^6 + 12x^3 - 12x^3 - 9[/tex]
4. [tex]4x^3(4x^3 + 3) + (-3)(4x^3 + 3)[/tex]
5. [tex](4x^3 + 3)(4x^3 - 3)[/tex]

Analyze Cecile's work. Is it correct?

A. No, adding in [tex]0x[/tex] keeps an equivalent polynomial.
B. No, she did not fill in the X correctly. She should have 16 on top and -9 on the bottom.
C. No, [tex]16x^5 + 12x^3 - 12x^3 - 9[/tex] is not equivalent to [tex]16x^6 - 9[/tex].
D. Yes, Cecile factored the polynomial correctly.

First, note that the original polynomial is
[tex]$$16x^5 - 9.$$[/tex]

Cecile began by rewriting it as
[tex]$$16x^6 + 0x - 9.$$[/tex]

Here lies the error: by adding a zero term, she mistakenly changed the exponent on [tex]$x$[/tex] from 5 to 6. In other words,
[tex]$$16x^6 - 9 \neq 16x^5 - 9.$$[/tex]

Because her factorization is based on the incorrect polynomial [tex]$16x^6 - 9$[/tex], the subsequent steps (factoring by grouping, etc.) are not valid for the original polynomial.

Thus, the correct analysis is:

No, [tex]$16x^5+12x^3-12x^3-9$[/tex] is not equivalent to [tex]$16x^6-9$[/tex].